Write $f ( x ) = x ^ { 5 } - 6 x ^ { 4 } + 15 x ^ { 3 } - 18 x ^ { 2 } + 10 x$ as a product of linear factors given that $2 - i$ and $1 - i$ are zeros of the polynomial.
A) $( x - 2 + i ) ^ { 3 } ( x - 1 - i ) ^ { 2 }$
B) $x ^ { 2 } ( x - 2 - i ) ( x - 1 + i ) ( x - 1 - i )$
C) $x ( x - 2 + i ) ( x - 2 - i ) ( x - 1 + i ) ( x - 1 - i )$
D) $x ^ { 3 } ( x - 2 - i ) ( x - 1 - i )$
E) $( x - 2 - i ) ^ { 2 } ( x - 1 - i ) ^ { 3 }$

Question

Quizplus · Accepted Answer

Given the complex roots $2 - i$ and $1 - i$, their conjugates $2 + i$ and $1 + i$ must also be roots due to the complex conjugate root theorem. Since the polynomial is of degree 5, and we have identified 4 roots (including their conjugates), the remaining factor must be linear and real, leading to the factor $x$. Thus, the correct factorization includes all these roots and the linear term $x$, making option C correct.

Write f(x)=x5−6x4+15x3−18x2+10xf ( x ) = x ^ { 5 } - 6 x ^ { 4 } + 15 x ^ { 3 } - 18 x ^ { 2 } + 10 xf(x)=x5−6x4+15x3−18x2+10x

Write $f ( x ) = x ^ { 5 } - 6 x ^ { 4 } + 15 x ^ { 3 } - 18 x ^ { 2 } + 10 x$