Write $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 16 x + 48$ as a product of linear factors.
A) $x = ( x - 3 ) ^ { 2 } ( x - 4 i )$
B) $x = ( x - 3 ) ( x - 4 ) ^ { 2 }$
C) $x = ( x + 3 ) ( x - 3 ) ( x + 4 )$
D) $x = ( x + 3 ) ( x + 4 i ) ( x - 4 i )$
E) $x = ( x - 3 ) ( x + 4 ) ^ { 2 }$

Question

Write $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 16 x + 48$ as a product of linear factors. 
A) $x = ( x - 3 ) ^ { 2 } ( x - 4 i )$
B) $x = ( x - 3 ) ( x - 4 ) ^ { 2 }$
C) $x = ( x + 3 ) ( x - 3 ) ( x + 4 )$
D) $x = ( x + 3 ) ( x + 4 i ) ( x - 4 i )$
E) $x = ( x - 3 ) ( x + 4 ) ^ { 2 }$

Quizplus · Accepted Answer

To factor $f(x) = x^3 + 3x^2 + 16x + 48$, we look for real roots and then consider complex roots if necessary. By trying factors of 48, we find that $x = -3$ is a root because $f(-3) = 0$. Factoring out $x + 3$, we get $f(x) = (x + 3)(x^2 + 16)$. The quadratic $x^2 + 16$ can't be factored over the real numbers, but it can be written as $(x + 4i)(x - 4i)$ using complex numbers. Therefore, the product of linear factors is $f(x) = (x + 3)(x + 4i)(x - 4i)$.

Write f(x)=x3+3x2+16x+48f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 16 x + 48f(x)=x3+3x2+16x+48

Write $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 16 x + 48$