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I need to find an nth degree polynomial function that has real coefficients using the following conditions:

n=3; 3 and 4i are zeros; f(2)=40

I have no idea what I'm doing on this one. It's been too long. Also, there's no homework tag because this isn't something I have to do. I'm just brushing up in preparation for an upcoming math course.

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Hint : Since $3$ and $4i$ are zeros, you can factorize your polynomial like this : $f(z)=(z-3)(z-4i)g(z)$. What is the degree of $g$ then ? –  Philippe Malot Apr 26 '13 at 9:50

1 Answer 1

up vote 2 down vote accepted

We want $3$ and $4i$ to be roots. If a real polynomial has a complex root, the conjugate also is a root. So we can just write down a polynomial that has exactly these roots: \begin{align*} f(x)=(x-3)(x-4i)(x+4i)=(x-3)(x^2+16) \end{align*} which indeed has only real coefficients and is of degree $3$. But the value at $2$ is not $40$ but $-20$ so the polynomial you are looking for is $g(x)=-2f(x)$.

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I need to practice some more of these, but you've pointed me in the right direction. Thanks. –  Dave Apr 26 '13 at 10:09

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