Write the polynomial in completely factored form. (Hint: One factor is $x ^ { 2 } + 1$.) $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + 14 x ^ { 2 } + 6 x + 13$
A) $f ( x ) = ( x - 2 ) ( x + 2 ) ( x + 3 - i ) ( x + 3 + i )$
B) $f ( x ) = ( x - 3 ) ( x + 3 ) ( x - 1 - 2 i ) ( x - 1 + 2 i )$
C) $f ( x ) = ( x - 1 ) ( x + 1 ) ( x + 3 - 2 i ) ( x + 3 + 2 i )$
D) $f ( x ) = ( x - 3 i ) ( x + 3 i ) ( x - 1 - 2 i ) ( x - 1 + 2 i )$
E) $f ( x ) = ( x - i ) ( x + i ) ( x + 3 - 2 i ) ( x + 3 + 2 i )$

Question

Quizplus · Accepted Answer

Given the hint that $x^2 + 1$ is a factor, which can be factored further into $(x - i)(x + i)$, we look for options that include complex conjugate pairs that simplify to $x^2 + 1$. Option E correctly represents this factorization, including the complex conjugate pair $(x + 3 - 2i)(x + 3 + 2i)$ that corresponds to the remaining factors of the polynomial when divided by $x^2 + 1$.

Write the Polynomial in Completely Factored Form x2+1x ^ { 2 } + 1x2+1

Write the Polynomial in Completely Factored Form $x ^ { 2 } + 1$