If you thought that adding and subtracting rational expressions was difficult, you are in for a nice surprise. Multiplying and dividing rational expressions is far easier. In fact, learning both multiplying and dividing rational expressions boils down to learning only how to multiply them. Division only involves one extra step.
Consider how we multiply regular fractions. We multiply straight across the top, straight across the bottom, then we simplify. East enough, right? For instance consider the problem
34×57
To multiply these two fractions, we simply multiply across the top, and across the bottom, like so
34×57=3⋅54⋅7=1528
The fraction 15/28 is already in simplest form, so we’re done. The procedure for multiplying rational expressions is exactly the same. Only the expressions are slightly more complex. Alright, let’s look at an example:
Multiply
p2+4p−5p+2×p+24p2−4p3
To solve this problem, we simply multiply across the top, and multiply across the bottom, then simplify.
p2+4p−5p+2×p+24p2−4p3=(p2+4p−5)(p+2)(p+2)(4p2−4p3)
=p2+4p−54p2−4p3, by cancelling p+2
=(p+5)(p−1)4p2(1−p), by factoring both numerator and denominator
=−p+54p2, since p−11−p=−1
So then, we have
p2+4p−5p+2×p+24p2−4p3=−p+54p2
So how does division of rational expressions work? Well, consider how division of fractions works. Since dividing fractions is akin to multiplying the first fraction by the inverse of the second, we use the simple technique of flipping the second fraction and then multiplying them! Easy as pie, right?
For instance, consider the problem
38÷23
Here dividing by 2/3 is the same thing as multiplying by 3/2 (since 2/3 and 3/2 are inverses of each other). So we simply flip the second fraction and multiply straight across.
38÷23=38×32=3⋅38⋅2=916
We use the same procedure to divide rational expressions. Consider the following problem.
Divide
4xx2−9x+8÷x−3−x2+9x−8
Simply flip the second rational expression, and then multiply.
4xx2−9x+8÷x−3−x2+9x−8=4xx2−9x+8×−x2+9x+8x−3
=(4x)(−x2+9x+8)(x2−9x+8)(x−3)
=−(4x)(x2−9x−8)(x2−9x+8)(x−3), by factoring-1 from −x2+9x+8
=−4xx−3, by cancelling x2−9x−8
Then our final result is
4xx2−9x+8÷x−3−x2+9x−8=−4xx−3
Below you can download some free math worksheets and practice.
Simplify each expression.
This free worksheet contains 10 assignments each with 24 questions with answers.
Example of one question:
Watch below how to solve this example:
Simplify each expression.
This free worksheet contains 10 assignments each with 24 questions with answers.
Example of one question:
Watch below how to solve this example:
Simplify each expression.
This free worksheet contains 10 assignments each with 24 questions with answers.
Example of one question:
Watch below how to solve this example: