Write $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x - 29$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6$ .
A) $f ( x ) = ( x - 6 ) \left( x ^ { 2 } - x + 4 \right)$
B) $f ( x ) = ( x - 6 ) \left( x ^ { 2 } - x + 4 \right) - 5$
C) $f ( x ) = ( x + 6 ) \left( x ^ { 2 } - x + 4 \right) + 5$
D) $f ( x ) = ( x + 6 ) \left( x ^ { 2 } + x + 4 \right) - 5$
E) $f ( x ) = ( x - 6 ) \left( x ^ { 2 } + x + 4 \right) + 5$

Question

Quizplus · Accepted Answer

To rewrite $f(x) = x^3 - 7x^2 + 10x - 29$ in the form $f(x) = (x - k)q(x) + r$ with $k = 6$, we perform polynomial division or use synthetic division with $k = 6$. The quotient $q(x)$ will be a polynomial of degree 2, and $r$ will be the remainder. After performing the division, we find that $f(x) = (x - 6)(x^2 - x + 4) - 5$, which matches choice B.

Write f(x)=x3−7x2+10x−29f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x - 29f(x)=x3−7x2+10x−29

Write $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x - 29$