Question

What is the complete factorization of the polynomial function over the set of complex numbers? f(x)=x3+3x2+16x+48

Answer 1

(x+3) (x-4i) (x+4i) is the answer, just took the quiz :)

Answer 2

The complete factorization of the given polynomial function is $f(x)=(x+2i)(x-2i)(x+3)$.

Given:

The given polynomial function is $f(x)=x^3+3x^2+16x+48$.

To find:

The complete factorization of the given polynomial function.

Explanation:

We have,

$f(x)=x^3+3x^2+16x+48$

By grouping, we get

$f(x)=(x^3+3x^2)+(16x+48)$

$f(x)=x^2(x+3)+16(x+3)$

$f(x)=(x^2+16)(x+3)$

It can be written as:

$f(x)=(x^2+4^2)(x+3)$

$f(x)=(x+2i)(x-2i)(x+3)$              $[\because a^2+b^2=(a+bi)(a-bi)]$

Therefore, the complete factorization of the given polynomial function is $f(x)=(x+2i)(x-2i)(x+3)$.

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