Answer
$f(x)=2x^{3}-8x^{2}+8x-32$
Work Step by Step
The Linear Factorization Theorem... in words:
An nth-degree polynomial can be expressed as the product of a nonzero constant and $n$ linear factors, where each linear factor has a leading coefficient of 1.
Imaginary roots, if they exist, occur in conjugate pairs.
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If $2i$ is a zero, so is $-2i.$
$f(x)=a_{n}(x-4)(x+2i)(x-2i)$
$f(x)=a_{n}(x-4)(x^{2}+4)$
Using the given $f(-1),$we find $a_{n}:$
$f(-1)=a_{n}(-1-4)(1+4)$
$-50=a_{n}(-25)$
$a_{n}=2$
so,
$f(x)=2(x-4)(x^{2}+4)$
$f(x)=2(x^{3}-4x^{2}+4x-16)$
$f(x)=2x^{3}-8x^{2}+8x-32$
