Question

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

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


NO GUESSING/RANDOM ANSWER, or reported

Answer 1

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

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.