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The following examples illustrate how to divide a polynomial by a monomial. The division process is quite simple and is based on addition of rational expressions.
$\frac{a}{c}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{b}{c}=\frac{a+b}{c}$
Turning this equation around we get
$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
Now we simply divide $c$ into $a$ , and $c$ into $b$ . This should suggest a rule.
$\begin{array}{l}\begin{array}{lll}\frac{3{x}^{2}+x-11}{x}.\hfill & \hfill & \text{Divideeverytermof}3{x}^{2}+x-11\text{by}x.\hfill \end{array}\hfill \\ \hfill \\ \frac{3{x}^{2}}{x}+\frac{x}{x}-\frac{11}{x}=3x+1-\frac{11}{x}\hfill \end{array}$
$\begin{array}{l}\begin{array}{lll}\frac{8{a}^{3}+4{a}^{2}-16a+9}{2{a}^{2}}.\hfill & \hfill & \text{Divideeverytermof}8{a}^{3}+4{a}^{2}-16a+9\text{by}2{a}^{2}.\hfill \end{array}\hfill \\ \hfill \\ \frac{8{a}^{3}}{2{a}^{2}}+\frac{4{a}^{2}}{2{a}^{2}}-\frac{16a}{2{a}^{2}}+\frac{9}{2{a}^{2}}=4a+2-\frac{8}{a}+\frac{9}{2{a}^{2}}\hfill \end{array}$
$\begin{array}{l}\begin{array}{lll}\frac{4{b}^{6}-9{b}^{4}-2b+5}{-4{b}^{2}}.\hfill & \hfill & \text{Divideeverytermof}\hfill \end{array}4{b}^{6}-9{b}^{4}-2b+5\text{\hspace{0.17em}}\text{by}-4{b}^{2}.\hfill \\ \hfill \\ \frac{4{b}^{6}}{-4{b}^{2}}-\frac{9{b}^{4}}{-4{b}^{2}}-\frac{2b}{-4{b}^{2}}+\frac{5}{-4{b}^{2}}=-{b}^{4}+\frac{9}{4}{b}^{2}+\frac{1}{2b}-\frac{5}{4{b}^{2}}\hfill \end{array}$
Perform the following divisions.
$\frac{2{x}^{2}+x-1}{x}$
$2x+1-\frac{1}{x}$
$\frac{3{x}^{3}+4{x}^{2}+10x-4}{{x}^{2}}$
$3x+4+\frac{10}{x}-\frac{4}{{x}^{2}}$
$\frac{{a}^{2}b+3a{b}^{2}+2b}{ab}$
$a+3b+\frac{2}{a}$
$\frac{14{x}^{2}{y}^{2}-7xy}{7xy}$
$2xy-1$
$\frac{10{m}^{3}{n}^{2}+15{m}^{2}{n}^{3}-20mn}{-5m}$
$-2{m}^{2}{n}^{2}-3m{n}^{3}+4n$
In Section [link] we studied the method of reducing rational expressions. For example, we observed how to reduce an expression such as
$\frac{{x}^{2}-2x-8}{{x}^{2}-3x-4}$
Our method was to factor both the numerator and denominator, then divide out common factors.
$\frac{\left(x-4\right)\left(x+2\right)}{\left(x-4\right)\left(x+1\right)}$
$\frac{\overline{)\left(x-4\right)}\left(x+2\right)}{\overline{)\left(x-4\right)}\left(x+1\right)}$
$\frac{x+2}{x+1}$
When the numerator and denominator have no factors in common, the division may still occur, but the process is a little more involved than merely factoring. The method of dividing one polynomial by another is much the same as that of dividing one number by another. First, we’ll review the steps in dividing numbers.
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