Write $f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 13 x + 22$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when k = -5.
A) $f ( x ) = ( x + 5 ) \left( x ^ { 2 } + 2 x + 3 \right) + 7$
B) $f ( x ) = ( x + 5 ) \left( x ^ { 2 } - 2 x + 3 \right) - 7$
C) $f ( x ) = ( x - 5 ) \left( x ^ { 2 } + 2 x + 3 \right) - 7$
D) $f ( x ) = ( x - 5 ) \left( x ^ { 2 } - 2 x + 3 \right) + 7$
E) $f ( x ) = ( x + 5 ) \left( x ^ { 2 } + 2 x + 3 \right)$

Question

Quizplus · Accepted Answer

To write $f(x) = x^3 + 7x^2 + 13x + 22$ in the form $f(x) = (x - k)q(x) + r$ with $k = -5$, we perform polynomial division or use synthetic division with $x = -5$. The quotient $q(x)$ is $x^2 + 2x + 3$ and the remainder $r$ is 7, resulting in $f(x) = (x + 5)(x^2 + 2x + 3) + 7$.

Write f(x)=x3+7x2+13x+22f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 13 x + 22f(x)=x3+7x2+13x+22

Write $f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 13 x + 22$