Multiplying and Dividing Rational Expressions: Practice Problems

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  • 0:05 Review
  • 0:24 Example #1
  • 1:44 Example #2
  • 3:14 Example #3
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Lesson Transcript
Instructor: Kathryn Maloney

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

Let's continue looking at multiplying and dividing rational polynomials. In this lesson, we will look at a couple longer problems, while giving you some practice multiplying and dividing.


Multiplication and division of rational polynomial expressions is easy once you remember the steps.

For multiplication: factor, cancel or slash, and multiply.

For division: factor, flip, cancel or slash, and multiply.

Let's do some larger problems.

Example #1

In example #1, cancel out the like terms to find the solution
Rational Polynomial Cancel Out Example

((q^2 - 11q + 24) / (q^2 - 18q + 80)) * ((q^2 - 15q + 50) / (q^2 - 9q + 20))

First, we need to factor. (q^2 - 11q + 24) factors into (q - 8)(q - 3). (q^2 - 18q + 80) factors into (q - 10)(q - 8). (q^2 - 15q + 50) factors into (q - 10)(q - 5). (q^2 - 9q + 20) factors into (q - 5)(q - 4).

So, this is what our new expression is going to look like: ((q - 8)(q - 3) / (q - 10)(q - 8)) * ((q - 10)(q - 5) / (q - 5)(q - 4))

Next, we are going to cancel (what I like to call slash) like terms. We're going to cancel or slash (q - 10) over (q - 10), (q - 8) over (q - 8), and finally (q - 5) over (q - 5).

Now that we have canceled or slashed all of the like terms from the top and bottom, we multiply straight across. Don't multiply anything we slashed because those are now 1's. It turns out, our answer is (q - 3) / (q - 4).

Example #2

In example #2, flip the second fraction before changing the problem to a multiplication one
Dividing Rational Expression Example 2

((y^2 - 9) / (2y + 1)) / ((3 - y) / (2y^2 + 7y + 3))

Let's factor. (y - 9) = (y - 3)(y + 3) and (2y^2 + 7y + 3) = (2y + 1)(y + 3). Our next step is to flip the second fraction and change it to multiplication. Our new expression is going to look like this: ((y - 3)(y + 3) / (2y + 1)) * ((2y + 1)(y + 3) / ((3 - y)).

The next step is canceling, or what we've been calling slashing. We can slash (2y + 1) over (2y + 1). In the numerator, we have (y - 3)(y + 3) and (y + 3). In the denominator we have (3 - y). If we multiply (3 - y) by -1, we'll get -1(y - 3). Guess what? We can cancel (y - 3) over (y - 3), but remember to leave the -1!

So, our final answer's going to look like: (y + 3)(y + 3) / -1.

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