Question:
Using the remainder theorem, find the remainder, when $p(x)$ is divided by $g(x)$, where $p(x)=2 x^{3}-9 x^{2}+x+15, g(x)=2 x-3 .$
Solution:
$p(x)=2 x^{3}-9 x^{2}+x+15$
$g(x)=2 x-3=2\left(x-\frac{3}{2}\right)$
By remainder theorem, when $p(x)$ is divided by $(2 x-3)$, then the remainder $=p\left(\frac{3}{2}\right)$.
Putting $x=\frac{3}{2}$ in $p(x)$, we get
$p\left(\frac{3}{2}\right)=2 \times\left(\frac{3}{2}\right)^{3}-9 \times\left(\frac{3}{2}\right)^{2}+\frac{3}{2}+15=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+15=\frac{27-81+6+60}{4}=\frac{12}{4}=3$
∴ Remainder = 3
Thus, the remainder when p(x) is divided by g(x) is 3.