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arsen [322]
3 years ago
9

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

Mathematics
1 answer:
earnstyle [38]3 years ago
5 0

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.

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