Question

Use synthetic division and the remainder theorem to find p(-2) if p(x)=x^(4)+2x^(3)-8x^(2)-18x-9

Answer 1

$p(x)=q(x)(x+2)+r(x)$

Notice that if $x=-2$, we're left with $p(-2)=r(-2)$.

Synthetic division yields

-2  |  1    2    -8    -18    -9
.    |       -2     0     16      4
- - - - - - - - - - - - - - - - - - -
.    |  1    0    -8      -2    -5

$\implies\dfrac{p(x)}{x+2}=\dfrac{x^4+2x^3-8x^2-18x-9}{x+2}=x^3+4x^2-18-\dfrac5{x+2}$
$\implies p(x)=(x^3+4x^2-18)(x+2)-5$
$\implies p(-2)=-5$

so the answer is (B).