Question

On dividing $3x^3+x^2+2x+5$ by a polynomial $g\left(x\right)$, the quotient and remainder are $(3x-5)$ and $(9x+10)$ respectively. Find $g\left(x\right)$

Answer 1

If we divide $f\left(x\right)=3x^3+x^2+2x+5$ by $g\left(x\right)$ we get $q\left(x\right)=(3x-5)$ as quotient and $r\left(x\right)=(9x+10)$ as remainder.

By using division algorithm,

$f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$

$⟹3x^3+x^2+2x+5=g\left(x\right)(3x-5)+(9x+10)$

$⟹g\left(x\right)=\frac{3x^3+x^2+2x+5-9x-10}{(3x-5)}$

$⟹g\left(x\right)=\frac{3x^3+x^2-7x-5}{(3x-5)}$

Hence, $g\left(x\right)=x^2+2x+1$