Dividing Rational Expressions
You probably know how to divide rational expressions containing monomials.
Now, we will use the same procedure to divide rational expressions that
contain binomials or trinomials.
Note:
Here is an example of dividing rational
expressions that contain monomials:
![](./articles_imgs/370/dividi1.gif)
Procedure —
To Divide Rational Expressions
Step 1 Invert the second fraction and change the division symbol,
÷, to the multiplication symbol,
·.
Step 2 Factor the numerators and denominators.
Step 3 Cancel all pairs of factors common to the numerators and
denominators.
Step 4 Multiply the numerators. Multiply the denominators.
We usually leave the answer in factored form.
Example 1
Find:
![](./articles_imgs/370/dividi2.gif)
Solution |
![](./articles_imgs/370/dividi3.gif) |
Step 1 Invert the second fraction
and change the division
symbol, ÷, to the
multiplication symbol, · . |
![](./articles_imgs/370/dividi4.gif) |
Step 2 Factor the numerators
and denominators.
|
![](./articles_imgs/370/dividi5.gif) |
Step 3 Cancel all pairs of factors
common to the numerators
and denominators.
|
![](./articles_imgs/370/dividi6.gif) |
Step 4 Multiply the numerators.
Multiply the denominators. |
![](./articles_imgs/370/dividi7.gif) |
Thus,
![](./articles_imgs/370/dividi8.gif)
Example 2
Find:
![](./articles_imgs/370/dividi9.gif)
Solution |
![](./articles_imgs/370/dividi10.gif) |
Step 1 Invert the second fraction
and change the division
symbol, ÷, to the
multiplication symbol, ·. |
![](./articles_imgs/370/dividi11.gif) |
Step 2 Factor the numerators and
denominators.
|
![](./articles_imgs/370/dividi12.gif) |
Step 3 Cancel all pairs of factors
common to the numerators and denominators. |
![](./articles_imgs/370/dividi13.gif) |
Replace
with
-1. |
![](./articles_imgs/370/dividi15.gif) |
Step 4 Multiply the numerators.
Multiply the denominators. |
![](./articles_imgs/370/dividi16.gif) |
Thus,
![](./articles_imgs/370/dividi17.gif)
Note:
You can also write the answer as follows:
![](./articles_imgs/370/dividi18.gif) |