Question:
Use remainder theorem to find the value of $k$, it being given that when $x^{3}+2 x^{2}+k x+3$ is divided by $(x-3)$, then the remainder is 21 .
Solution:
Let $p(x)=x^{3}+2 x^{2}+k x+3$
Now, $p(3)=(3)^{3}+2(3)^{2}+3 k+3$
$=27+18+3 k+3$
$=48+3 k$
It is given that the remainder is 21
$\therefore 3 k+48=21$
$\Rightarrow 3 k=-27$
$\Rightarrow k=-9$