Question

$f\left(x\right)$ is a polynomial when divided by $(x-2)$ and $(x+2)$ leaves remainder $5$ and $1$ respectively and when divided by $(x^2-4)$ leaves $g\left(x\right)$ as remainder, then $g\left(5\right)$ is equal to

Answer 1

Given : $f\left(x\right)$ is a polynomial when divided by $(x-2)$ and $(x+2)$ leaves remainder $5$ and $1.$
$\Rightarrow f\left(2\right)=5,f(-2)=1$ and when divided by $(x^2-4)$ leaves $g\left(x\right)$
$\Rightarrow f\left(x\right)=(x^2-4)Q\left(x\right)+ax+b\Rightarrow f\left(2\right)=2a+b=5\cdots \left(1\right)\Rightarrow f(-2)=-2a+b=1\cdots \left(2\right)$
Solving equations $\left(1\right)$ and $\left(2\right),$ we get
$\Rightarrow a=1,b=3\Rightarrow g\left(x\right)=x+3\therefore g\left(5\right)=8$