Polynomials and Factoring

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.

Example.

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 $–x+4$.

Mini-Lecture.

Below is a mini lecture about long division for polynomials.

Example.

Below is an applet to practice long division for polynomials.


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