Diferència de quadrats |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨24px¨»«mrow»«mfenced mathcolor=¨#191919¨»«mrow»«mi mathcolor=¨#0000FF¨»a«/mi»«mo»+«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«/mrow»«/mfenced»«mo mathcolor=¨#191919¨»§#183;«/mo»«mfenced mathcolor=¨#191919¨»«mrow»«mi mathcolor=¨#0000FF¨»a«/mi»«mo»-«/mo»«mi mathcolor=¨#FF0000¨»b«/mi»«/mrow»«/mfenced»«mo mathcolor=¨#191919¨»§#160;«/mo»«mo mathcolor=¨#191919¨»=«/mo»«mo mathcolor=¨#191919¨»§#160;«/mo»«msup mathcolor=¨#191919¨»«mi mathcolor=¨#0000FF¨»a«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#191919¨»-«/mo»«msup mathcolor=¨#191919¨»«mi mathcolor=¨#FF0000¨»b«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» Exemples:
(4x + 9)(4x - 9) = (4x)2 - (9)2 = 16x2 - 81 (2x3 + 5y)(2x3 - 5y) = (2x3)2 - (5y)2 = 4x6 - 25y2 |
Quadrat d'una suma/diferència |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨24px¨»«mrow»«msup»«mfenced»«mrow»«mi mathvariant=¨normal¨ mathcolor=¨#0000FF¨»a«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FF0000¨»b«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi mathvariant=¨normal¨ mathcolor=¨#0000FF¨»a«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#0000FF¨»a«/mi»«mo»§#183;«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FF0000¨»b«/mi»«mo»+«/mo»«msup»«mi mathvariant=¨normal¨ mathcolor=¨#FF0000¨»b«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math»
Exemple:
(4x + 9)2 = (4x)2 + 2·4x·9 + (9)2 = 16x2 + 72x + 81 Però si és una diferència: (4x – 9)2 = (4x + (-9))2 = 16x2 – 72x + 81 |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»A«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»A«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/math»
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨»x«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mstyle»«/math»
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»A«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»A«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»B«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/math»
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨»x«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mstyle»«/math»
]]>Treure factor comú |
Per treure factor comú a una expressió algèbrica, cal calcular el mcd dels sumands.
Exemple:
18x5 - 9x4 + 36x3 - 27x2 = 9x2(2x3 - x2 + 4x - 3) El quadrat d'un binomi i la diferència de quadrats poden ser útils.
Treure factor comú pot ajudar a simplificar:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»3«/mn»«mi»x«/mi»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#160;«/mo»«/math»
|
Però també es pot identificar un quadrat perfecte,
si es treu #a1 en factor: #a1 · (#e1);
s'identifica el doble producte 2·#a2 = #p
i el quadrat del segon (#a2)2 = #q
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Però també es pot identificar un quadrat perfecte,
si es treu #a1 en factor: #a1 · (#e1);
s'identifica el doble producte 2·#a2 = #p
i el quadrat del segon (#a2)2 = #q
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Però també es pot identificar una diferència de quadrats,
si es treu #a1 en factor: #a1 · (#e1);
es veu que #q = #a22
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Signe de ax + b | |||
x | -∞ | -b/a | +∞ |
ax+b | Signe contrari de a | 0 | Signe de a |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»e«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mfrac mathcolor=¨#007F00¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«/math»
Després cal fer la taula de signes.
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x | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mo»§#8734;«/mo»«/math» | #s |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«mo»§#8734;«/mo»«/math»
|
#e | #s_1 | 0 | #s_2 |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»e«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mfrac mathcolor=¨#007F00¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«/math»
Després cal fer la taula de signes.
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x | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mo»§#8734;«/mo»«/math» | #s |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«mo»§#8734;«/mo»«/math»
|
#e | #s_1 | 0 | #s_2 |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»e«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8660;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mfrac mathcolor=¨#007F00¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«/math»
Després cal fer la taula de signes.
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x | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mo»§#8734;«/mo»«/math» | #s |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«mo»§#8734;«/mo»«/math»
|
#e | #s_1 | 0 | #s_2 |
Signe de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msup mathcolor=¨#FFFFC3¨»«mi mathcolor=¨#FFFFC3¨»ax«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathcolor=¨#FFFFC3¨»bx«/mi»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»c«/mi»«/mstyle»«/math» |
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Si no té solucions, té el signe de a. Si té solució doble x1, té el signe de a excepte en x1 Si té dues solucions, x1< x2: |
|||||
x | -∞ | x1 | x2 | +∞ | |
a | Signe de a | 0 | Signe de a | 0 | Signe de a |
x-x1 | — | 0 | + | + | |
x-x2 | — | — | 0 | + | |
ax2+bx+c | Signe de a | Contrari de a | Signe de a |
Factoritza: 2(x-1)(x-2)
Resol:
el discriminant és #d i les solucions són #x_1 i #x_2.
ES COMPROVEN AMB S = #S1 I P = #P1
Es pot factoritzar amb a·(x – x1)(x – x2). No oblidis el signe de a
Després es fa la taula.
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Com que a és #r1, #e és #r2, #r4, per x<#x_1 o per x>#x_2 #e és #r3, #r5, per #x_1<x i per x<#x_2 |
|||||
x | –∞ | #x_1 | #x_2 | +∞ | |
a | #s3 | #s3 | #s3 | ||
#k11 | — | 0 | + | + | |
#k12 | — | — | 0 | + | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Factoritza: 2(x-1)(x-2)
Resol:
el discriminant és #d i les solucions són #x_1 i #x_2.
ES COMPROVEN AMB S = #S1 I P = #P1
Es pot factoritzar amb a·(x – x1)(x – x2). No et deixis a
Després es fa la taula.
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Com que a és #r1, #e és #r2, #r4, per x<#x_1 o per x>#x_2 #e és #r3, #r5, per #x_1<x i per x<#x_2 |
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x | –∞ | #x_1 | #x_2 | +∞ | |
a | #s3 | #s3 | #s3 | ||
#k11 | — | 0 | + | + | |
#k12 | — | — | 0 | + | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Signe de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨22px¨»«mrow»«msup mathcolor=¨#FFFFC3¨»«mi mathcolor=¨#FFFFC3¨»ax«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathcolor=¨#FFFFC3¨»bx«/mi»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»c«/mi»«/mrow»«/mstyle»«/math» | |||||
Si no té solució, té el signe de a. Si té solució doble, té el signe de a excepte en x1. Si té dues solucions, x1 i x2: |
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x | -∞ | x1 | x2 | +∞ | |
ax2+bx+c | Signe de a | 0 | Contrari de a | 0 | Signe de a |
Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mi»x«/mi»«mo»§lt;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altra situació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>el discriminant és #d i les solucions són #x_1 i #x_2.
(ES COMPROVEN AMB S = #S1 I P = #P1)
Després faig la taula.
]]>
Com que a és #r1, |
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x | -∞ | #x_1 | #x_2 | +∞ | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mi»x«/mi»«mo»§lt;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altra situació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>el discriminant és #d i les solucions són #x_1 i #x_2.
(ES COMPROVEN AMB S = #S1 I P = #P1)
Després es fa la taula.
]]>
Com que a és #r1, #e és #r2 (signe de a), #r4, per x<#x_1 o per x>#x_2 #e és #r3 (signe contrari de a), #r5, per #x_1<x i per x<#x_2 |
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x | -∞ | #x_1 | #x_2 | +∞ | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»2«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mn»5«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altres situacions: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>el discriminant és #d i les solucions són #x_1 i #x_2.
ES COMPROVEN AMB S = #S1 I P = #P1
Després es fa la taula.
]]>
Com que a és #r1, #e és #r2 (signe de a), #r4, per x<#x_1 o per x>#x_2 #e és #r3 (signe contrari de a), #r5, per #x_1<x i per x<#x_2 |
|||||
x | -∞ | #x_1 | #x_2 | +∞ | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»2«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mn»5«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altres situacions: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>el discriminant és #d i les solucions són #x_1 i #x_2.
ES COMPROVEN AMB S = #S1 I P = #P1
Després es fa la taula.
]]>
Com que a és #r1, #e és #r2 (signe de a), #r4, per x<#x_1 o per x>#x_2 #e és #r3 (signe contrari de a), #r5, per #x_1<x i per x<#x_2 |
|||||
x | -∞ | #x_1 | #x_2 | +∞ | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mn»2«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mn»5«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altres situacions: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>el discriminant és #d i les solucions són #x_1 i #x_2.
ES COMPROVEN AMB S = #S1 I P = #P1
Després es fa la taula.
]]>
Com que a és #r1, #e és #r2 (signe de a), #r4, per x<#x_1 o per x>#x_2 #e és #r3 (signe contrari de a), #r5, per #x_1<x i per x<#x_2 |
|||||
x | -∞ | #x_1 | #x_2 | +∞ | |
#e | #s1 | 0 | #s2 | 0 | #s1 |
Signe d'una fracció |
Recorda que: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mrow»«mfrac mathcolor=¨#003300¨»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»+«/mo»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»-«/mo»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#003300¨»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»-«/mo»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»+«/mo»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«/mrow»«/mstyle»«/math» |
Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mi»x«/mi»«mo»§lt;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altra situació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>x | -∞ | #x_1 | #x_2 | +∞ | |
#n | #d3 | 0 | #d4 | #d5 | |
#d | #d6 | #d7 | 0 | #d8 | |
#e | #s_1 | 0 | #s_2 | #s_1 |
i l'enunciat demana que l'expressió sigui #sig
]]>Entre dos nombres: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§lt;«/mo»«mi»x«/mi»«mo»§#8743;«/mo»«mi»x«/mi»«mo»§lt;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8743;«/mo»«mo»=«/mo»«mi»i«/mi»«mo»)«/mo»«/math»
Altra situació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§lt;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#8744;«/mo»«mi»x«/mi»«mo»§#62;«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mo»§#8744;«/mo»«mo»=«/mo»«mi»o«/mi»«mo»)«/mo»«/math»
]]>x | -∞ | #x_1 | #x_2 | +∞ | |
#n | #d3 | 0 | #d4 | #d5 | |
#d | #d6 | #d7 | 0 | #d8 | |
#e | #s_1 | 0 | #s_2 | #s_1 |
i l'enunciat demana que l'expressió sigui #sig
]]>Addició de polinomis |
Per sumar polinomis, sumem els monomis de mateix grau. Exemple: (-2x4 + 7x3+ 6x2 + 3x - 1) + (4x3- 6x2 + 9x - 5) = -2x4 + 11x3 + 12x - 6 Perquè:
-2x4 + 0 = -2x4 7x3 + 4x3 = 11x3 6x2+ (- 6x2) = 0x2(que no s'escriu) 3x + 9x = 12x - 1+(-5) = -6 |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»b«/mi»«mn mathvariant=¨bold¨»4«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»5«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math»
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»4«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»3«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»b«/mi»«mn mathvariant=¨bold¨»4«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»5«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math»
]]>ÉS MILLOR SI SUMEU LES FRACCIONS SENSE CALCULADORA
]]>Subtracció de polinomis |
Per restar el polinomi Q(x) del polinomi P(x), restem dels monomis de P(x) els monomis de mateix grau de Q(x). Exemple: (-2x4 + 7x3+ 6x2 + 3x - 1) - (-5x4- 6x2 + 3x - 5) = 3x4 + 7x3+ 12x2+ 4 Perquè:
-2x4 - (-5x4) = 3x4 7x3 - 0= 7x3 6x2- (- 6x2) = 12x2 3x - 3x = 0x (que no s'escriu) - 1-(-5) = 4 |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/mrow»«/mstyle»«/math»
2. Resta = suma aquest oposat
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»
També ho pots fer directament
]]>ÉS MILLOR SI RESTEULES FRACCIONS SENSE CALCULADORA
]]>Multiplicació per un monomi |
Per a multiplicar el polinomi P(x) pel monomi Q(x) multipliquem TOTS els monomis de P(x) per Q(x). Agrupem i ordenem. Exemple: (-2x2)·(7x3+ 2x - 1) = (-2x2)· 7x3 + (-2x2)·2x + (-2x2)·(-1) = -14x5- 4x3+ 2x2 |
#a · (#b) =
(#a1·x#n) · (#b1·x4)
+ (#a1·x#n) · (#b2·x3)
+ (#a1·x#n) · (#b3·x2)
+ (#a1·x#n) · (#b4·x)
+ (#a1·x#n) · (#b5)
]]>(#a1·x#n) · (#b1·x4) = #c1 · x#n1
+ (#a1·x#n) · (#b2·x3) = #c2 · x#n2
+ (#a1·x#n) · (#b3·x2) = #c3 ·x#n3
+ (#a1·x#n) · (#b4·x) = #c4 · x#n4
+ (#a1·x#n) · (#b5) = #c5 · x#n5
]]>Multiplicació de 2 polinomis |
Per a multiplicar el polinomi P(x) pel polinomi Q(x) multipliquem TOTS els monomis de P(x) per TOTS els monomis de Q(x). Agrupem i ordenem. Exemple: (-2x2+3x)·(7x3+ 2x - 1)= (-2x2)· 7x3 + (-2x2)·2x + (-2x2)·(-1) + 3x·7x3 + 3x·2x + 3x·(- 1) = -14x5- 4x3+ 2x2 + 21x3+ 6x2- 3x Agrupat i ordenat: -14x5 + 17x3 + 8x2 - 3x
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a_«/mi»«mn mathvariant=¨bold¨»5«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/mfenced»«/mstyle»«/math»
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»
Ara ordena i agrupa els monomis per graus.
]]>
Operacions combinades |
Quan cal efectuar operacions combinades, cal tenir present
|
Després es fa la #r11: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sig«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»sig«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»qr«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mstyle»«/math»
]]>Després es fa la resta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mi mathvariant=¨bold¨»Q«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»p«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»qr«/mi»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»
]]>Després es fa la #w1: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sig«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mi mathvariant=¨bold¨»Q«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»p«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sig«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»
]]>Després es fa el producte: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨[¨ close=¨]¨»«mrow»«mi mathvariant=¨bold¨»Q«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»p«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»qr«/mi»«/mrow»«/mfenced»«/mstyle»«/math»
]]>després el producte com a diferència de quadrats: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»r«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»qr«/mi»«/mstyle»«/math»
Després es fa la resta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow mathcolor=¨#0000FF¨»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»qr«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mstyle»«/math»
]]>després la potència com un producte: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»q«/mi»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»q«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«/mstyle»«/math»
Després es fa la resta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow mathcolor=¨#0000FF¨»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mstyle»«/math»
]]>després, la diferència de quadrats: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»p«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»12«/mn»«/mstyle»«/math»
Després es fa la resta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»12«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»qr«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mstyle»«/math»
]]>Binomi de Newton |
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El coeficient que procedeix del triangle de Pascal és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«/mrow»«/mstyle»«/math».
Cal elevar «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#183;«/mo»«msup mathcolor=¨#000066¨»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»x«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»t«/mi»«/mrow»«/msup»«/mrow»«/mstyle»«/math» a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#000066¨»2«/mn»«/mrow»«/mstyle»«/math» i «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«msup mathcolor=¨#000066¨»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»x«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mrow»«/msup»«/mrow»«/mstyle»«/math» a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#000066¨»3«/mn»«/mrow»«/mstyle»«/math»
]]>Format de la resposta:
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Format de la resposta:
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Format de la resposta:
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Format de la resposta:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨normal¨»Q«/mi»«mo»=«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»5«/mn»«msup»«mi mathvariant=¨normal¨»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»x«/mi»«mo»-«/mo»«mn»7«/mn»«mspace linebreak=¨newline¨/»«mi mathvariant=¨normal¨»R«/mi»«mo»=«/mo»«mo»-«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math»
Quin és el polinomi P(x)?
Format de la resposta:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»5«/mn»«msup»«mi mathvariant=¨normal¨»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»x«/mi»«mo»-«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math»
Dividend = quocient · divisor + residu
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Valor numèric d'un polinomi |
El valor numèric d'un polinomi s'obté substituint la indeterminada per un nombre concret. Exemple: si P(x)=3x3 - 5x2 + 8x - 1, P(-1) = 3·(-1)3 - 5·(-1)2 + 8·(-1) -1 = 3·(-1) - 5·1 + 8·(-1) -1 = -17 P(1) = 3 ·13 - 5 · 12 + 8 · 1 - 1 = 3 - 5 + 8 - 1 = 5 |
També pots calcular el residu de la divisió per #w1
]]>Teorema del residu |
El residu de la divisió d'un polinomi P(x) per x - a és igual al valor numèric del polinomi per x = a o sigui: residu = P(a) |
També es pot calcular el valor numèric per x = #a, i dona P(#a) = #r1
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Factoritzar polinomi sense terme independent |
Si un polinomi no té terme independent, es treu x en factor amb el màxim grau possible. Exemple: -2x4 + 7x3+ 6x2 = x2 · (-2x2 + 7x + 6) |
x#n ·(#Q)
#Q no es pot factoritzar perquè té un discriminant negatiu.
]]>x#n ·(#Q)
#Q no es pot factoritzar ja que té discriminant negatiu.
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Factoritzar polinomi amb identitat remarcable |
Abans de començar càlculs val la pena comprovar si el polinomi no es pot escriure com una identitat remarcable. -2x2 + 50 = -2·(x2 - 25) = -2(x+5)(x-5) |
#P = #a · (#P1)
]]>#P = #a · (#P1)
]]>#P = #a · (#P1)
]]>#P = #a · (#P1)
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Factoritzar polinomi 2n grau amb solucions |
Si un polinomi de grau 2 té solucions, es pot escriure: ax2 + bx + c = a (x - x1)(x - x2) Atenció: no et deixis la a! |
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]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#916;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«msup mathcolor=¨#0000FF¨»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»d«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#177;«/mo»«msqrt»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«/msqrt»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8660;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨{¨ close=¨}¨»«mtable»«mtr»«mtd»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»La«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»factoritzaci§#243;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#233;s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»ax«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»bx«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»NO«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»OBLIDIS«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»DE«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»POSAR«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»LA«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»a«/mi»«/math»
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Factoritzar un polinomi amb la regla de Ruffini |
Si un polinomi té terme independent i grau superior a 2, es va dividint per (x - a) fins al grau 2 pel mètode de Ruffini. Exemple: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msup mathcolor=¨#000033¨»«mi mathcolor=¨#000033¨»x«/mi»«mn»4«/mn»«/msup»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»5«/mn»«msup mathcolor=¨#000033¨»«mi mathcolor=¨#000033¨»x«/mi»«mn»3«/mn»«/msup»«mo mathcolor=¨#000033¨»+«/mo»«mn mathcolor=¨#000033¨»20«/mn»«mi mathcolor=¨#000033¨»x«/mi»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»16«/mn»«/mrow»«/mstyle»«/math». Ruffini dona com arrels 1 i 2. El quocient de 2n grau té per solucions: -2 i 4: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msup mathcolor=¨#000033¨»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mn»4«/mn»«/msup»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»5«/mn»«msup mathcolor=¨#000033¨»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mn»3«/mn»«/msup»«mo mathcolor=¨#000033¨»+«/mo»«mn mathcolor=¨#000033¨»20«/mn»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»16«/mn»«mo mathcolor=¨#000033¨»=«/mo»«mo mathcolor=¨#000033¨»(«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mo mathcolor=¨#000033¨»+«/mo»«mn mathcolor=¨#000033¨»2«/mn»«mo mathcolor=¨#000033¨»)«/mo»«mo mathcolor=¨#000033¨»(«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»1«/mn»«mo mathcolor=¨#000033¨»)«/mo»«mo mathcolor=¨#000033¨»(«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»2«/mn»«mo mathcolor=¨#000033¨»)«/mo»«mo mathcolor=¨#000033¨»(«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#000033¨»x«/mi»«mo mathcolor=¨#000033¨»-«/mo»«mn mathcolor=¨#000033¨»4«/mn»«mo mathcolor=¨#000033¨»)«/mo»«/mrow»«/mstyle»«/math» |
El primer divisor és #t1, i el quocient #P1.
]]>#M2 #M3 #M4
]]>El primer divisor és #t1:
#c4 | #c3 | #c2 | #c1 | #c0 | |
#f1 | |||||
El segon divisor és #t2:
#c4 | #c3 | #c2 | #c1 | #c0 | |
#f1 | #h3 | #h2 | #h1 | #h0 | |
#g3 | #g2 | #g1 | #g0 | 0 | |
#f2 |
Ara cal resoldre, si es pot, l'equació de 2n grau.
]]>El primer divisor és #d2
]]>En el quocient, es pot treure #a en factor comú per posar en evidència la diferència de quadrats:
#P1 = #a · (#P2)
]]>En el quocient, es pot treure #a en factor comú per posar en evidència el quadrat d'una suma:
#P1 = #a · (#P2)
]]>Després fins a grau 2, s'aplica el mètode de Ruffini.
L'últim quocient de grau 2 és una diferència de quadrats.
]]>Després fins a grau 2, s'aplica el mètode de Ruffini.
L'últim quocient de grau 2 és una diferència de quadrats.
]]>a) 4(x+2)(x-2)
b) {-1,1,2,2} Posa-les totes, si n'hi ha de dobles, escriu-les dos cops
]]>Es fa el canvi de variable t = x2 i t2 = x4, i es resol per trobar t.
El pas següent és trobar x desfent el canvi de variable amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#177;«/mo»«msqrt mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨»t«/mi»«/msqrt»«/mrow»«/mstyle»«/math»
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Mètode de Ruffini amb arrels fraccionàries |
Si un polinomi té terme independent i grau superior a 2, es va dividint per (x-a) fins al grau 2 pel mètode de Ruffini. Si el coeficient del terme de grau més alt és diferent de 1, pot ser que a sigui una fracció. DIVISORS FRACCIONARIS: Són totes les fraccions que es poden formar amb al numerador els divisors del terme independent i al denominador els divisors del coeficient del terme de grau més alt . Amb 6x5 - 3x4 + 7x3 +2x+15 cal provar ±1/2,±1/3,±1/5,±3/2... |
Com que el coeficient del grau més alt, no és 1, es proven les fraccions formades:
Arrel d'un polinomi |
a és arrel d'un polinomi P(x) si, i només si, P(a) = 0. Per determinar les arrels d'un polinomi de grau superior a 2 es factoritza, si el grau és inferior o igual a 2, es resol l'equació polinòmica següent. i les arrels són 0 (doble), 2 i 3. Es diu que a és arrel múltiple d'un polinomi si en la factorització, el factor (x-a) apareix més d'un cop. k és la multiplicitat de l'arrel a si el factor corresponent és (x-a)k |
El primer divisor és #t1, i el quocient #P1.
Per factoritzar #P1, #M.
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Format de la resposta: {-1,2,2,4}
#P és un polinomi de grau 4, #M1.
#M2 #M3 #M4
]]>#P és un polinomi de grau 4, #M1.
#M2 #M3 #M4
]]>Ara ja es poden trobar les arrels del polinomi de 2n grau que queda, trobant les solucions de l'equació de segon grau corresponent:
]]>Després aplica el mètode de Ruffini al polinomi de grau 3 que has trobat.
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El quocient del mètode de Ruffini és un polinomi de 2n grau amb dues solucions, la 4a i la 5a arrel.
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mcm i mcd de diferents polinomis |
El mcm de diferents polinomis es calcula multiplicant entre ells tots els factors comuns i no comuns elevats a l'exponent màxim de les factoritzacions. El mcd de diferents polinomis es calcula multiplicant entre ells exclusivament els factors comuns elevats a l'exponent mínim de les factoritzacions. |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/mrow»«/mstyle»«/math»
només cal escollir els factors comuns amb l'exponent més petit.
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/mrow»«/mstyle»«/math»
només cal agafar els factors comuns amb l'exponent més petit
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/mrow»«/mstyle»«/math»
Agafa TOTS els factors amb l'exponent més GRAN
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Q«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/mrow»«/mstyle»«/math»
Agafa TOTS factors amb l'exponent més GRAN
]]>Simplifica fraccions algèbriques |
Per simplificar fraccions algèbriques, dividim el seu numerador i el seu denominador per el mcd d'ambdós. A la practica, "tatxem" els factors repetits al numerador i al denominador si TOTS els factors s'estant multiplicant entre ells; MAI si s'estan sumant.
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»P«/mi»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»Q«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»P«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»Q«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/math»
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»M«/mi»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»N«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
sabent que «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mstyle»«/math» són arrels del numerador i del denominador.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»M«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»N«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Ja pots simplificar
]]>Primer cal reduir-les al mínim comú denominador que és el polinomi mcm dels denominadors (Tots els factors amb l'exponent més gran)
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#191919¨»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»5«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»+«/mo»«mfrac mathcolor=¨#191919¨»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»25«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»=«/mo»«mfrac mathcolor=¨#191919¨»«mrow»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathcolor=¨#0000FF¨ mathvariant=¨bold¨»(«/mo»«mi mathcolor=¨#0000FF¨ mathvariant=¨bold¨»x«/mi»«mo mathcolor=¨#0000FF¨ mathvariant=¨bold¨»+«/mo»«mn mathcolor=¨#0000FF¨ mathvariant=¨bold¨»5«/mn»«mo mathcolor=¨#0000FF¨ mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathcolor=¨#0000FF¨ mathvariant=¨bold¨»(«/mo»«mi mathcolor=¨#0000FF¨ mathvariant=¨bold¨»x«/mi»«mo mathcolor=¨#0000FF¨ mathvariant=¨bold¨»+«/mo»«mn mathcolor=¨#0000FF¨ mathvariant=¨bold¨»5«/mn»«mo mathcolor=¨#0000FF¨ mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»+«/mo»«mfrac mathcolor=¨#191919¨»«mrow»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»x«/mi»«/mrow»«mrow»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Quan tenen el mateix denominador, es sumen els numeradors.
L'expressió es transforma en:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/math»
Ara ja tenen el mateix denominador i es poden sumar els numeradors:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»4«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«/math»
]]>el denominador comú per la suma és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»k«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»11«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨».«/mo»«/mrow»«/mstyle»«/math»
L'expressió es transforma en:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»k«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»k«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»k«/mi»«mn mathvariant=¨bold¨»12«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»k«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Ara ja tenen el mateix denominador i es pot calcular el numerador, multiplicant i sumant.
]]>
amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«/mrow»«/mstyle»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«/mrow»«/mstyle»«/math»
]]>i que #d2 = #d21,
el mcm dels denominadors és #mc11.
]]>
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»mc«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»mc«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=e99ead11a57c6034b2d2d5fd8c01bfbd&cw=303&ch=27&cb=18" style="vertical-align: -9px;"/>
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»11«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»fn«/mi»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»12«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»fd«/mi»«mspace linebreak=¨newline¨/»«/mrow»«/mstyle»«/math»
el seu mcm, i per tant el denominador comú és:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»w«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/mrow»«/mstyle»«/math»
]]>i «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»21«/mn»«/mrow»«/mstyle»«/math»,
el denominador comú és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»mc«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»11«/mn»«/mrow»«/mstyle»«/math».
]]>la suma es transforma en:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»t«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#FF0000¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»)«/mo»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#FF0000¨»3«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»t«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#FF0000¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»t«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»t«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#FF0000¨»3«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»t«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»t«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#FF0000¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»)«/mo»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
]]>amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
]]>L'expressió es transforma en:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced mathcolor=¨#FF0000¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Ara ja tenen el mateix denominador i es poden sumar els numeradors:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»-«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»4«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«/mrow»«/mstyle»«/math»
]]>amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«/mrow»«/mstyle»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«/mrow»«/mstyle»«/math»
]]>i que #d2 = #d21,
el mcm dels denominadors és #mc11.
]]>
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»mc«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»mc«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«/mrow»«/mstyle»«/math»
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»11«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»fn«/mi»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»12«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»fd«/mi»«mspace linebreak=¨newline¨/»«/mrow»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=1edb4609d30bbc06c8d62d7a287ecfb9&cw=84&ch=27&cb=22" style="vertical-align: -5px;"/>
el seu mcm, i per tant el denominador comú és:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»w«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/mrow»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=28a71b18b9f3136cf9717240d2845cd5&cw=31&ch=10&cb=9" style="vertical-align: -1px;"/>
]]>Es presenta el numerador de la 1a multiplicat pel numerador de la 2a i el denominador de la 1a multiplicat pel denominador de la 2a:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac mathcolor=¨#003300¨»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#183;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mstyle»«/math»
Es factoritza el numerador i el denominador per simplificar:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»x«/mi»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«/mstyle»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»G«/mi»«/mrow»«/mstyle»«/math»
Resultat simplificat i factoritzat
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
b) Es factoritza el numerador: #f1
es factoritza el denominador: #f2
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
i es simplifica pel seu mcd que és #mc1
]]>Resultat simplificat
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»F«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»G«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/math»
b) Es factoritza el numerador: #f1
es factoritza el denominador: #f2
i es simplifica pel seu mcd que és #mc1
]]>Es multiplica la 1a per la inversa de la segona:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#003300¨»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#183;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mstyle»«/math»
Es factoritza el numerador i el denominador per simplificar:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»x«/mi»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«/mstyle»«/math»
Resultat simplificat i factoritzat
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»F«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Després es factoritza
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»F«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfrac»«/mstyle»«/math»
i es simplifica
]]>«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»F«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mi mathvariant=¨bold¨»G«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Després es factoritza i es simplifica:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfrac»«/mstyle»«/math»
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