factor the expression completely over the complex numbers. $x^{3}-4x^{2}+4x - 16$ enter your answer in the…

Explanation:

Step1: Group the terms

$(x^{3}-4x^{2})+(4x - 16)$

Step2: Factor out common factors from each group

$x^{2}(x - 4)+4(x - 4)$

Step3: Factor out the common binomial factor

$(x - 4)(x^{2}+4)$

Step4: Factor $x^{2}+4$ over complex numbers

Since $x^{2}+4=x^{2}-(- 4)=x^{2}-(2i)^{2}$, and using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$, we have $x^{2}+4=(x + 2i)(x - 2i)$. So the completely factored form is $(x - 4)(x + 2i)(x - 2i)$.

Answer:

$(x - 4)(x + 2i)(x - 2i)$