Dividing Polynomials

Example 1:

Divide.

$\left(15p^{2}q+25p{q}^2-10pq\right)÷5pq$

Write as a rational expression, and factor the numerator.

$\frac{15p^{2}q+25p{q}^2-10pq}{5pq}=\frac{5pq\left(3p+5q-2\right)}{5pq}$

Cancel the common factor.

$=3p+5q-2$

Example 2:

Divide.

$\left(m^2+4m-12\right)÷\left(m+3\right)$

We can factor the dividend as $\left(m+6\right)\left(m-2\right)$ , but that doesn't help us in this case. Instead, we'll use long division.

Write using the long division symbol. Divide the leading terms, $m^2÷m$

$\begin{array}{l}m\\ m+3\overline{)m^2+4m-12}\end{array}$

Multiply and subtract.

$\begin{array}{l}m\hfill \\ m+3\overline{)m^2+4m-12}\hfill \\ \underset{\_}{m^2+3m}\hfill \\ m\hfill \end{array}$

Bring down the $-12$ . Again, divide the leading terms, $m÷m$ .

$\begin{array}{l}m+1\hfill \\ m+3\overline{)m^2+4m-12}\hfill \\ \underset{\_}{m^2+3m}\hfill \\ m-12\hfill \end{array}$

Multiply and subtract.

$\begin{array}{l}m+1\hfill \\ m+3\overline{)m^2+4m-12}\hfill \\ \underset{\_}{m^2+3m}\hfill \\ m-12\hfill \\ \underset{\_}{m+3}\hfill \\ -15\hfill \end{array}$

Write the quotient with remainder.

$\left(m^2÷m+4m-12\right)÷\left(m+3\right)=m+1-\frac{15}{m+3}$