Department of Mathematics

Polynomials and Factoring

Worked Examples

Dividing Polynomials (Long Division)

Dividing polynomials using long division is analogous to dividing numbers. To compute $32/11$, for instance, we ask how many times $11$ fits into $32$. Similarly, we start dividing polynomials by seeing how many times one leading term fits into the other.


Compare $32\div 11=2+\frac{10}{11}$ and $$ (x^3+x^2-2x+3)\div(x^2-1)=x+1 + \frac{-x+4}{x^2-1}.$$
$ x^3 + x^2 - 2x +3$ $\qquad$ $ |\underline{ \ x^2-1 \quad }$ $\qquad$
$- \quad$ $\underline{x^3 \phantom{+x^2}-\phantom{2}x \phantom{+3}}$ $\pmb{x}$ (Details)Comparing leading terms, $x^2$ needs to be multiplied by $x$ to get $x^3$, so $x$ is the first term in our answer.
$ \phantom{x^3+} x^2 - \phantom{2}x +3$ (Details)We subtract $x(x^2-1)$ from the polynomial we are dividing.
$-$ $ \phantom{x^3+} \underline{x^2 \phantom{-2x} -1}$ $ \phantom{\pmb{x}} \mathbf{+1}$ (Details)Comparing leading terms of the new polynomial, $x^2-x+3$, and $x^2-1$: $x^2$ fits into $x^2$ once, so the next term in the answer is $1$.
$ \phantom{x^3+ x^2} - \phantom{2}x +4 $ (Details) We subtract $1(x^2-1)$.
We stop here because the leading term of $x^2-1$, i.e. $x^2$, cannot be fitted into the leading term of $–x+4$, i.e. $–x$.
So, we get a remainder of $–x+4$.


Below is a mini lecture about long division for polynomials.


Below is an applet to practice long division for polynomials.

Note: There are two main styles of long division. The above example is usually referred to as the "European style". If it looks unfamiliar (i.e., you are used to the "radical sign" style), make sure to check the reference below for other examples.

Reference: See also the Wikipedia article on polynomial long division.