Write $f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6 + \sqrt { 6 }$.
A) $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$
B) $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] + 5$
C) $f ( x ) = [ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] + 5$
D) $f ( x ) = [ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$
E) $f ( x ) = [ x + ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } - ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5$

Question

Quizplus · Accepted Answer

We can use synthetic division to find the remainder, but first we need to rewrite the function in terms of $x-k$. We have:
\begin{align*}
x-k&=x-(6+\sqrt{6})\
&=x-6-\sqrt{6}\
&=(x-6)-\sqrt{6}
\end{align*}
So we can substitute $x-6-\sqrt{6}$ for $x$ in $f(x)$ to get:
\begin{align*}
f(x)&=(x-6-\sqrt{6})^3-11(x-6-\sqrt{6})^2+18(x-6-\sqrt{6})+25\
&=((x-6)-\sqrt{6})^3-11((x-6)-\sqrt{6})^2+18((x-6)-\sqrt{6})+25\
&=[(x-6)^3-3\cdot(\sqrt{6})(x-6)^2+3\cdot6(x-6)-\sqrt{6}^3]\
&\qquad-11[(x-6)^2-2(\sqrt{6})(x-6)+\sqrt{6}^2]+18(x-6)-18\sqrt{6}+25\
&=-x^3+3(6+\sqrt{6})x^2+[9\sqrt{6}-18(6+\sqrt{6})]x-[-5+\sqrt{6}+11(6^2-6\sqrt{6}+\sqrt{6}^2)]\
&=-x^3+3(6+\sqrt{6})x^2+[9\sqrt{6}-18(6+\sqrt{6})]x-5.
\end{align*}
Using synthetic division with $6+\sqrt{6}$, we get:
$$
\begin{array}{c|cccc}
6+\sqrt{6} & -1 & 3(6+\sqrt{6}) & 9\sqrt{6}-18(6+\sqrt{6}) & -5 \ \hline
 & & -\sqrt{6} & 5+6\sqrt{6} & -5 \
\end{array}
$$
So we have $f(x)=q(x)(x-(6+\sqrt{6}))+r(x)$, where $q(x)=x^2+(5-\sqrt{6})x+(6-\sqrt{6})$ and $r(x)=-\sqrt{6}x+(5+6\sqrt{6})$.

Therefore, the correct answer is $\boxed{\textbf{(D) }f(x)=[ x - ( 6 + \sqrt { 6 } ) ] \left[ x ^ { 2 } + ( - 5 + \sqrt { 6 } ) x - ( 6 - \sqrt { 6 } ) \right] - 5}.$

Write f(x)=x3−11x2+18x+25f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25f(x)=x3−11x2+18x+25

Write $f ( x ) = x ^ { 3 } - 11 x ^ { 2 } + 18 x + 25$