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

Since the coefficients of the polynomial are real and $2i$ is a zero, its complex conjugate $-2i$ must also be a zero due to the Complex Conjugate Root Theorem. Option C correctly includes both $2i$ and its conjugate $-2i$ as roots, and the remaining real root $-1$ to ensure the polynomial is factored correctly given the information.

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