Answer:
[tex]f(x) =( {x} - 4i)(x + 4i)(x + 3)[/tex]
Step-by-step explanation:
The given polynomial is
[tex]f(x) = {x}^{3} + 3 {x}^{2} + 16x + 48[/tex]
Let us factor by grouping:
[tex]f(x) = {x}^{2} (x + 3) + 16(x + 3)[/tex]
We factor further to get:
[tex]f(x) = ( {x}^{2} + 16)(x + 3)[/tex]
We need to get the quadratic term factored.
[tex]f(x) =( {x}^{2} - ( - {4}^{2} ))(x + 3)[/tex]
[tex]f(x) =( {x}^{2} - ( {4i)}^{2} )(x + 3)[/tex]
We apply difference of two squares to get:
[tex]f(x) =( {x} - 4i)(x + 4i)(x + 3)[/tex]
This is the completely factored form over the complex numbers.
