Question

Divide $3x^5-8x^4-5x^3+26x^2-33x+26$ by $x^3-2x^2-4x+8$.

Answer 1

We know that $p\left(x\right)=q\left(x\right)\times g\left(x\right)+r\left(x\right)$ , where $q\left(x\right)$ and $r\left(x\right)$ are the quotient and remainder respectively when polynomial $p\left(x\right)$ is divided by polynomial $g\left(x\right)$

Also, degree of $r\left(x\right)<$ degree of $g\left(x\right)$ and degree of $q\left(x\right)$ = degree of $p\left(x\right)-$ degree of $g\left(x\right)$

Given $p\left(x\right)=3x^5-8x^4-5x^3+26x^2-33x+26$ and $g\left(x\right)=x^3-2x^2-4x+8$

Degree of $q\left(x\right)$ is 2, therefore assuming $q\left(x\right)=A{x}^2+B{x}^{+}C$

Coefficient of $x^5$ in $g\left(x\right)\times q\left(x\right)$ = coefficient of $x^5$ in $p\left(x\right)[\because p(x)=q(x)\times g(x)+r(x\left)\right]$

$⟹A=3$

Similarly equating the coefficients of $x^4$ and $x^3$, we get

$-2A+B=-8⟹B=-2$ and $-4A-2B+C=-5⟹C=3$

$\therefore q\left(x\right)=3x^2-2x+3$

$q\left(x\right)\times g\left(x\right)=3x^5-8x^4-5x^3+26x^2-28x+24$

and $r\left(x\right)=p\left(x\right)-q\left(x\right)\times g\left(x\right)=-5x+2$