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How do you write a polynomial function of least degree that has real coefficients, the following given zeros 2,-2,-6i and a leading coefficient of 1?

1 Answer

May 27, 2016

$f (x) = x^{4} + 32 x^{2} - 144$ $f(x) = x^4+32x^2-144$

Explanation:

Since we want Real coefficients, any non-Real zeros must occur in Complex conjugate pairs.

So both $- 6 i$ $-6i$ and $6 i$ $6i$ are zeros and the simplest polynomial with these zeros is:

$f (x) = (x - 2) (x + 2) (x - 6 i) (x + 6 i)$ $f(x) = (x-2)(x+2)(x-6i)(x+6i)$

$= (x^{2} - 2^{2}) (x^{2} - {(6 i)}^{2})$ $= (x^2-2^2)(x^2-(6i)^2)$

$= (x^{2} - 4) (x^{2} + 36)$ $= (x^2-4)(x^2+36)$

$= x^{4} + 32 x^{2} - 144$ $= x^4+32x^2-144$

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