SOLUTION: How do you find an nth-degree polynomial function with real coefficients satisfying the given conditions? n=3; 1 and 5i are zeros; f (-1)=-104

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Question 316701: How do you find an nth-degree polynomial function with real coefficients satisfying the given conditions? n=3; 1 and 5i are zeros; f (-1)=-104
Answer by solver91311(24713) About Me  (Show Source):
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Use the following facts:

The Fundamental Theorem of Algebra, namely that every -th degree polynomial function has exactly zeros, counting all multiplicities.

Complex roots ALWAYS come in conjugate pairs. That means that if is a zero, then is also a zero of the desired polynomial function.

If is a zero of a polynomial function in , then is a factor of the polynomial.

A family of polynomial functions of the form:



all have the same zeros.

So we know the following things:

1. The desired polynomial function has exactly 3 zeros.

2. The zeros are , , and .

3. The factors of the polynomial, including any possible common factor multiplier are:



Multiply the factors:





Since



Hence and





John