# Verify the division algorithm for the polynomials

Question:

Verify the division algorithm for the polynomials $p(x)=2 x^{4}-6 x^{3}+2 x^{2}-x+2$ and $g(x)=x+2$.

Solution:

$p(x)=2 x^{4}-6 x^{3}+2 x^{2}-x+2$ and $g(x)=x+2$

Quotient $=2 x^{3}-10 x^{2}+22 x-45$

Remainder = 92

Verification:

Divisor $\times$ Quotient + Remainder

$=(x+2) \times\left(2 x^{3}-10 x^{2}+22 x-45\right)+92$

$=x\left(2 x^{3}-10 x^{2}+22 x-45\right)+2\left(2 x^{3}-10 x^{2}+22 x-45\right)+92$

$=2 x^{4}-10 x^{3}+22 x^{2}-45 x+4 x^{3}-20 x^{2}+44 x-90+92$

$=2 x^{4}-6 x^{3}+2 x^{2}-x+2$

= Dividend

Hence verified.