Question

4 points
write a linear factorization of the function.
$f(x) = x^3 + 2x^2 + 16x + 32$
$f(x) = (x + 4)(x + 2i)(x - 2i)$
$f(x) = (x + 2)(x + 4i)(x - 4i)$
$f(x) = (x + 2i)(x + 4i)(x - 4i)$
$f(x) = (x + 4i)(x + 2i)(x - 2i)$
clear my selection

Explanation:

Step1: Factor by grouping

Group the terms of \( f(x)=x^{3}+2x^{2}+16x + 32\):
\( (x^{3}+2x^{2})+(16x + 32) \)
Factor out common factors from each group:
\( x^{2}(x + 2)+16(x + 2) \)
Now factor out \( (x + 2) \):
\( (x + 2)(x^{2}+16) \)

Step2: Factor the quadratic

Recall that \( a^{2}+b^{2}=(a + bi)(a - bi) \). For \( x^{2}+16\), we have \( a = x \) and \( b = 4 \), so:
\( x^{2}+16=(x + 4i)(x - 4i) \)

Step3: Combine factors

Putting it all together, the linear factorization of \( f(x) \) is:
\( f(x)=(x + 2)(x + 4i)(x - 4i) \)

Answer:

\( f(x)=(x + 2)(x + 4i)(x - 4i) \) (the second option: \( f(x)=(x + 2)(x + 4i)(x - 4i) \))