To factorize the given polynomial [tex]\( f(x) = x^3 - 4x^2 + 4x - 16 \)[/tex] over the set of complex numbers, follow these steps:
1. Identify the polynomial: The polynomial in question is [tex]\( f(x) = x^3 - 4x^2 + 4x - 16 \)[/tex].
2. Factor the polynomial: Break down the polynomial into its simplest factors over the set of complex numbers.
The polynomial can be factorized as:
[tex]\[
f(x) = (x - 4)(x^2 + 4)
\][/tex]
3. Verify the factorization: To ensure the correctness of the factorization, we can multiply the factors back together:
\begin{align}
(x - 4)(x^2 + 4) &= x \cdot x^2 + x \cdot 4 - 4 \cdot x^2 - 4 \cdot 4 \\
&= x^3 + 4x - 4x^2 - 16 \\
&= x^3 - 4x^2 + 4x - 16
\end{align}
As we can see, multiplying these factors returns the original polynomial.
Thus, the complete factorization of the polynomial function [tex]\( f(x) = x^3 - 4x^2 + 4x - 16 \)[/tex] over the set of complex numbers is:
[tex]\[
(x - 4)(x^2 + 4)
\][/tex]
So, the answer to this problem, written in the box provided, would be:
[tex]\[
(x - 4)(x^2 + 4)
\][/tex]
