# How can we find the quotient and remainder when: f(x)=x^5−x^4−4x^3+2x+3 is divided by g(x)=x−2?

How can we find the quotient and remainder when:
$f\left(x\right)={x}^{5}-{x}^{4}-4{x}^{3}+2x+3$
is divided by
$g\left(x\right)=x-2?$
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A practical method is as follow:
$\begin{array}{r}\\ {x}^{5}-{x}^{4}-4{x}^{3}+2x+3& & & & & x-2\\ {x}^{4}-4{x}^{3}+2x+3& & & & & {x}^{4}\\ -2{x}^{3}+2x+3& & & & & {x}^{3}\\ -4{x}^{2}+2x+3& & & & & -2{x}^{2}\\ -6x+3& & & & & -4x\\ -9& & & & & -6\end{array}$
so the quotient is ${x}^{4}+{x}^{3}-2{x}^{2}-4x-6$ and the remainder is -9.
and to explain the procedure of calculus: we divide the leading term ${x}^{5}$ of the dividend by the leading term x of the divisor we find ${x}^{4}$ and then we calculate:
${x}^{5}-{x}^{4}-4{x}^{3}+2x+3-{x}^{4}\left(x-2\right)={x}^{4}-4{x}^{3}+2x+3=R\left(x\right)$
and repeat the same calculus using R(x) as your new dividend until we find the remainder R(x) with degree less than the degree of the divisor x-2.