Question

(30 POINTS HELP) What is the complete factorization of the polynomial function over the set of complex numbers?

Answer 1

Answer:

$f(x) =( {x} - 4i)(x + 4i)(x + 3)$

Step-by-step explanation:

The given polynomial is

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

Let us factor by grouping:

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

We factor further to get:

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

We need to get the quadratic term factored.

$f(x) =( {x}^{2} - ( - {4}^{2} ))(x + 3)$

$f(x) =( {x}^{2} - ( {4i)}^{2} )(x + 3)$

We apply difference of two squares to get:

$f(x) =( {x} - 4i)(x + 4i)(x + 3)$

This is the completely factored form over the complex numbers.