Pre-Calc/Chapter 2: Polynomial, Power, and Rational Functions/2.5 Complex Zeros of Polynomials/2.5.1 Fundamental Theorem of Algebra
Match the polynomial function graph to the appropriate zeros and multiplicities.
Match the polynomial function graph to the appropriate zeros and multiplicities.
]]>
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1.0000000
0.3333333
0
true
true
abc
2 (multiplicity 2), -4 (multiplicity 2)
Incorrect
2 (multiplicity 2), -4 (multiplicity 3)
Correct
2 (multiplicity 3), -4 (multiplicity 3)
Incorrect
2 (multiplicity 3), -4 (multiplicity 2)
Incorrect
State how many complex and real zeros the function has.f = x4 - 63x2 - 64
State how many complex and real zeros the function has.
f = x4 - 63x2 - 64]]>
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1.0000000
0.3333333
0
true
true
abc
4 complex zeros; 1 real
Incorrect
4 complex zeros; all 4 real
Incorrect
4 complex zeros; none real
Incorrect
4 complex zeros; 2 real
Correct
State how many complex and real zeros the function has.f = x5 - 9x4 + 2x3 - ...
State how many complex and real zeros the function has.
f = x5 - 9x4 + 2x3 - 18x2 + x - 9]]>
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
1.0000000
0.3333333
0
true
true
abc
5 complex zeros; none real
Incorrect
5 complex zeros; 1 real
Correct
5 complex zeros; all 5 real
Incorrect
5 complex zeros; 3 real
Incorrect
Pre-Calc/Chapter 2: Polynomial, Power, and Rational Functions/2.5 Complex Zeros of Polynomials/2.5.2 Finding Complex Zeros
Using the given zero, find all other zeros of f(x).-2i is a zero of f(x) = x4...
Using the given zero, find all other zeros of f(x).
-2i is a zero of f(x) = x4 - 12x2 - 64]]>
1.0000000
0.3333333
0
true
true
abc
2i, 8, -8
Incorrect
2i, 4i, -4i
Incorrect
2i, 4, -4
Correct
2i, 8i, -8i
Incorrect
Write a linear factorization of the function.f(x) = 4x3 + 4x
Write a linear factorization of the function.
f(x) = 4x3 + 4x]]>
1.0000000
0.3333333
0
true
true
abc
f(x) = 4x(x + i)(x - i)
Correct
f(x) = x(x + 4i)(x - i)
Incorrect
2]]>
Incorrect
2]]>
Incorrect
Write a linear factorization of the function.f(x) = x3 + 8x2 + 16x + 128
Write a linear factorization of the function.
f(x) = x3 + 8x2 + 16x + 128]]>
1.0000000
0.3333333
0
true
true
abc
f(x) = (x + 4)(x + 8i)(x - 8i)
Incorrect
f(x) = (x + 8)(x + 4i)(x - 4i)
Correct
f(x) = (x + 4i)(x + 8i)(x - 8i)
Incorrect
f(x) = (x + 8i)(x + 4i)(x - 4i)
Incorrect
Write a polynomial function of minimum degree with real coefficients whose ...
Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
3 (multiplicity 1) and -2 (multiplicity 3)]]>
1.0000000
0.3333333
0
true
true
abc
= x4 + 3x3 - 6x2 + 28x - 24]]>
/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAARABEDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD7b+D/AMQtB0r4f+EviF4svJLfXfigLW+GoTWs0sdtHc5m0/THuFQxwxwR3CwIW8pZZTJJtEtxJud4h8a2nhPxN4b+JPhm0v7/AMJ+KNVt/DniGCC3NqJZ7meG00/V9txs8wJII7YvEuZoLqKTfIlpCtZPw0+Gi/Eb9nTwj8Mdbvlt9N8LWSeE/FukLCwuLqeyhSFfLlDq0EMhSK7jZk3ywS27YjEhBs+KPDvirW38A/Ci48U23iLU7HU7HxRrviCfTCsiaXp99DPaxSKkxC3lzPDEiyMFjkS3v3VA0QjKV9b/AC/r+v8AMfS39f1/W+n0HRRRTA8q8Of8nT/EP/sTPDX/AKXa7R8G/wDkovx2/wCxztv/AFHtGoooA9VooooA/9k=
Incorrect
= x4 + 3x3 - 6x2 - 28x + 24]]>
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
Incorrect
= x4 + 3x3 - 6x2 - 28x - 24]]>
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
Correct
= x4 - 3x3 - 6x2 - 28x - 24]]>
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
Incorrect
Write the function as a product of linear and irreducible quadratic factors, ...
Write the function as a product of linear and irreducible quadratic factors, all with real coefficients.
f = x3 - 2x2 + 2x - 15]]>
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
1.0000000
0.3333333
0
true
true
abc
]]>
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Incorrect
]]>
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Incorrect
]]>
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Correct
]]>
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Incorrect
Write the polynomial in standard form and identify the zeros of the function....
Write the polynomial in standard form and identify the zeros of the function.
f(x) = (x - 5i)(x +5i)]]>
1.0000000
0.3333333
0
true
true
abc
2 + 5ix + 25; zeros ± 5]]>
Incorrect
2 - 25; zeros ± 5i]]>
Incorrect
2 + 25; zeros ± 5]]>
Incorrect
2 + 25; zeros ± 5i]]>
Correct