Question

Verify the division algorithm for the polynomials

$p\left(x\right)=2x^4-6x^3+2x^2-x+2$ and $g\left(x\right)=x+2$.

Answer 1

Step $1:$ Dividing $p\left(x\right)=2x^4-6x^3+2x^2-x+2$ from $g\left(x\right)=x+2$:

We get, Quotient $=2x^3-10x^2+22x-45$ and Remainder $=92$

Step $2:$ Verifying the division algorithim:

$\mathrm{Dividend}=\mathrm{Quotient}\times \mathrm{Divisor}+\mathrm{Remainder}$

Taking R.H.S., we get,

$\begin{array}{rcl}\mathrm{Quotient}\times \mathrm{Divisor}+\mathrm{Remainder}& =& \left(2{\mathrm{x}}^3-10{\mathrm{x}}^2+22\mathrm{x}-45\right)\times \left(\mathrm{x}+2\right)+92\\ & =& \left(2{\mathrm{x}}^3-10{\mathrm{x}}^2+22\mathrm{x}-45\right)\times \mathrm{x}+\left(2{\mathrm{x}}^3-10{\mathrm{x}}^2+22\mathrm{x}-45\right)\times 2+92\\ & =& \left(2{\mathrm{x}}^4-10{\mathrm{x}}^3+22{\mathrm{x}}^2-45\mathrm{x}\right)+\left(4{\mathrm{x}}^3-20{\mathrm{x}}^2+44\mathrm{x}-90\right)+92\\ & =& 2{\mathrm{x}}^4-10{\mathrm{x}}^3+22{\mathrm{x}}^2-45\mathrm{x}+4{\mathrm{x}}^3-20{\mathrm{x}}^2+44\mathrm{x}-90+92\\ & =& 2{\mathrm{x}}^4-10{\mathrm{x}}^3+4{\mathrm{x}}^3+22{\mathrm{x}}^2-20{\mathrm{x}}^2-45\mathrm{x}+44\mathrm{x}-90+92\\ & =& 2{\mathrm{x}}^4-6{\mathrm{x}}^3+2{\mathrm{x}}^2-\mathrm{x}+2\\ & =& \mathrm{Dividend}\end{array}$

Hence, the division algorithm is verified for given polynomials.