Use synthetic division to express $P ( x ) = 3 x ^ { 3 } - 16 x ^ { 2 } - 7 x - 55$ in the form $($ divisor) $($ quotient $) +$ remainder for the divisor $x - 6$.
A) $\quad ( x - 6 ) \left( 3 x ^ { 2 } + 5 \right) - 25$
B) $( x - 3 ) \left( 25 x ^ { 2 } + 2 x + 5 \right) - 25$
C) $( x - 6 ) \left( 3 x ^ { 2 } + 2 x + 5 \right) - 25$
D) $( x - 6 ) \left( 3 x ^ { 2 } + x + 5 \right)$
E) none of these

Question

Quizplus · Accepted Answer

We can use synthetic division as follows:

\begin{array}{c|rrr|r}
6 & 3 & -16 & -7 & -55 \
\hline
& & 18 & 15 & 72 \
\hline
3 & -16 & 11 & 8 & 17 \
\end{array}

Thus, we have:

$$P(x) = (x-6)(3x^2+18x+15) + 17 = (x-6)(3x^2+2x+5) - 25.$$

Therefore, the correct choice is $\boxed{\textbf{(C)}}$.

Use Synthetic Division to Express P(x)=3x3−16x2−7x−55P ( x ) = 3 x ^ { 3 } - 16 x ^ { 2 } - 7 x - 55P(x)=3x3−16x2−7x−55

Use Synthetic Division to Express $P ( x ) = 3 x ^ { 3 } - 16 x ^ { 2 } - 7 x - 55$