Write $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 16$ as a product of linear factors.
A) $x = ( x - 4 ) ( x - 2 ) ^ { 2 }$
B) $x = ( x - 4 ) ( x + 2 i ) ( x - 2 i )$
C) $x = ( x - 4 ) ^ { 2 } ( x - 2 i )$
D) $x = ( x - 4 ) ( x + 2 ) ^ { 2 }$
E) $x = ( x + 4 ) ( x - 4 ) ( x + 2 )$

Question

Write $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 16$ as a product of linear factors. 
A) $x = ( x - 4 ) ( x - 2 ) ^ { 2 }$
B) $x = ( x - 4 ) ( x + 2 i ) ( x - 2 i )$
C) $x = ( x - 4 ) ^ { 2 } ( x - 2 i )$
D) $x = ( x - 4 ) ( x + 2 ) ^ { 2 }$
E) $x = ( x + 4 ) ( x - 4 ) ( x + 2 )$

Quizplus · Accepted Answer

By using synthetic division or the factor theorem, one can quickly determine that (x-4) is a factor of f(x). After dividing f(x) by (x-4), we are left with a quadratic that can be factored using the quadratic formula or by completing the square to obtain (x+2i)(x-2i). Therefore, the factorization of f(x) is (x-4)(x+2i)(x-2i), which corresponds to Choice B.

Write f(x)=x3−4x2+4x−16f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 16f(x)=x3−4x2+4x−16

Write $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 16$