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 $f(x)$, and since the coefficients of $f(x)$ are real, the complex conjugate of $k$, which is $-2i$, must also be a zero of $f(x)$. This means one of the factors of $f(x)$ is $(x - 2i)$. The remaining factor must be a quadratic polynomial with real coefficients, since the original polynomial has real coefficients. Among the given options, only option D correctly includes the factor $(x - 2i)$ and results in a polynomial with all real coefficients when expanded.

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