Factor f(x) into linear factors given that k is a zero of f(x).
-$f ( x ) = x ^ { 3 } - ( 4 + 2 i ) x ^ { 2 } + ( - 5 + 8 i ) x + 10 i ; k = 2 i$
A) $f ( x ) = ( x - i ) ( x + 1 ) ( x - 10 )$
B) $f ( x ) = ( x - 2 i ) ( x - 1 ) ( x + 5 )$
C) $f ( x ) = ( x + 2 i ) ( x - 1 ) ( x - 5 )$
D) $f ( x ) = ( x - 2 i ) ( x + 1 ) ( x - 5 )$

Question

Quizplus · Accepted Answer

Given that $k = 2i$ is a zero of $f(x)$, one of the factors must be $(x - 2i)$. The coefficients of $f(x)$ are complex numbers, so the complex conjugate root theorem states that if $2i$ is a root, then its complex conjugate $-2i$ must also be a root, unless all coefficients are real, which they are not in this case. However, none of the options include $(x + 2i)$, indicating that the question assumes only $2i$ as the given complex root without requiring its conjugate as a factor in the options. Therefore, we look for an option that includes $(x - 2i)$ as a factor. Option D is the correct one because it includes $(x - 2i)$ and is the only option that correctly incorporates the given zero $2i$. The other factors $(x + 1)$ and $(x - 5)$ are determined by further factoring the polynomial after dividing by $(x - 2i)$, but the key point for the selection is the inclusion of the given zero.

Factor F(x) into Linear Factors Given That K Is a Zero