Question

Find the divisor $g\left(x\right)$, when the polynomial $p\left(x\right)=4x^3+2x^2-10x+2$ is divided by $g\left(x\right)$ and the quotient and remainder obtained are $(2x^2+4x+1)$ and $5$ respectively

Answer 1

We know that the division algorithm states that:

Dividend$=($Divisor$\times$quotient$)+$Remainder

Here it is given that the dividend is $p\left(x\right)=4x^3+2x^2-10x+2$, the divisor is $g\left(x\right)$, the quotient is $2x^2+4x+1$ and the remainder is $5$, therefore,

$4x^3+2x^2-10x+2=\left[g\right(x\left)\right(2x^2+4x+1\left)\right]+5\Rightarrow g\left(x\right)=\frac{4x^3+2x^2-10x+2}{2x^2+4x+1}$

The division is shown above.

Hence, from the above division, we get that the divisor is $g\left(x\right)=2x-3$