Question

Polynomial $P ( x )$ contains only terms of odd degree. When $P ( x )$ is divided by $( x -3)$, the remainder is 6. If $P ( x )$ is divided by $\left( x ^2-9\right)$ then remainder is $g ( x )$. Find the value of $g (2)$.

Answer 1

As $P(x)$ is an odd function
hence $P(-x)=-P(x) \Rightarrow P(-3)=-P(3)=-6$
let $ P ( x )= Q \left( x ^2-9\right)+ ax + b $ (where $Q$ is quotient and $( ax + b )= g ( x )=$ remainder)
now $P (3)=3 a + b =6$....(1)
$P(-3)=-3 a+b=-6$.....(2)
hence $b=0$ and $a=2$
hence $g(x)=2 x \Rightarrow g(2)=4$