Question

On dividing the polynomial $3x^3+4x^2+5x-13$ by a polynomial $g\left(x\right)$, the quotient and the remainder were $(3x+10)$ and $(16x-43)$ respectively. Find $g\left(x\right)$.

Answer 1

Given that,

let Polynomial $p\left(x\right)=3x^3+4x^2+5x-13$

quotient $g\left(x\right)=3x+10$

remainder $r\left(x\right)=16x-43$

$g\left(x\right)=?$

Now,

we know that

Euclid division lemma theorem,

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

$3x^3+4x^2+5x-13=g\left(x\right)\times (3x+10)+(16x-43)$

$3x^3+4x^2+5x-13-16x+43=g\left(x\right)\times (3x+10)$

$3x^3+4x^2-11x+30=g\left(x\right)\times (3x+10)$

$g\left(x\right)=\frac{3x^3+4x^2-11x+30}{3x+10}$

now, dividing,

$3x+10)3x^3+4x^2-11x+30(x^2-2x+3-3x^3+10x^2\_\_\_\_\_\_\_\_\_\_\_\_\_-6x^2-11x-6x^2-20x\_\_\_\_\_\_\_\_\_\_\_\_\_9x+309x+30\_\_\_\_\_\_\_\_\_\_\_\_0$

Hence, $g\left(x\right)=x^2-2x+3$

This is the answer.