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 one factor is $x^2 + 1$, which can be factored further into $(x - i)(x + i)$, we look for a pair of complex conjugates that, when multiplied, give a quadratic with real coefficients. The correct option, E, provides the factor $x^2 + 1$ as $(x - i)(x + i)$ and another pair of complex conjugates $(x + 3 - 2i)(x + 3 + 2i)$ that also yields a quadratic with real coefficients when multiplied. This matches the structure of the given polynomial and its hint.

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

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