# Polynomials: Quotient and Remainder [duplicate]

How can we find the quotient and remainder when:

$$f(x)= x^5-x^4-4x^3+2x+3$$

is divided by $$g(x)=x-2?$$

Could someone please show how to step-by-step using synthetic division?

## marked as duplicate by jaykirby, Amzoti, Dan Rust, Start wearing purple, user67258 Aug 8 '13 at 13:33

A practical method is as follow:

\begin{align}\\ x^5-x^4-4x^3+2x+3&&&&& x-2\\ x^4-4x^3+2x+3 &&&&&x^4\\ -2x^3+2x+3 &&&&& x^3\\ -4x^2+2x+3 &&&&&-2x^2\\ -6x+3 &&&&&-4x\\ -9 &&&&&-6 \end{align} so the quotient is $x^4+x^3-2x^2-4x-6$ 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: $$x^5-x^4-4x^3+2x+3-x^4(x-2)=x^4-4x^3+2x+3=R(x)$$

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 $x-2$.

I still want you to attempt the question since it is really a matter of applying the synthetic division. Where do you fail to understand it? See the example below where $f(x)=x^{5}-2x^{3}-3x^{2}$ and $g(x)=x-1$. Follow the steps

(1) If it is not clear, ask where it is unclear