
Hi, and welcome to this video about rational expressions.
Reviewing Terminology
Before we talk about what rational expressions are and the operations that can be performed with them, it may be a good idea to review some terminology.
Polynomials
A polynomial is a group of algebraic or numeric terms that are joined by the operations of addition or subtraction. There are different types of polynomials based on the number of terms that are present:
Types of Polynomials
Examples | # of Terms | Type |
---|---|---|
1 | Monomial | |
2 | Binomial | |
3 | Trinomial |
Rational Expressions
A rational expression is nothing more than a ratio of polynomials. As you know from previous practice with ratios, you cannot divide by 0. It is important to keep this in mind when dealing with rational expressions because allowing a value of 0 in the denominator would create an expression that is “undefined.”
Using function notation for polynomials, such as
Here’s an example:
This example shows a rational expression with a monomial,
Addition and Subtraction of Rational Expressions
Rational expressions cannot be added or subtracted unless they share a common denominator. Algebraic rules allow us to adjust fractions to create common denominators as long as we make the same adjustment to the numerator. Let’s look at an example with fractions:
In order to add
We need to multiply the first denominator of the first expression by 7 to get to 35, but we also must multiply the numerator by the same value. Because we have simply created an equivalent fraction to allow us to add. Likewise, the second expression must be multiplied by
Rational expressions are added and subtracted the same way. Typically, the expressions need to be factored before the least common denominator can be determined and domain restrictions (excluded values) should be noted. Consider this example:
Now we want to determine the lowest common denominator. What is the smallest multiple of
Alright now that we have our equations written out, we want to make sure that we don’t have any domains that need to be excluded. Which, we do. Remember we don’t want 0 in the denominator position. So, in these scenarios, we know that
Alright, now we need to adjust the first expression by multiplying by the factor needed to match the least common denominator. So if we want our first term here, to match this term over here in the denominator position, we’re going to multiply by
Now we’re going to rewrite the expression as a fraction, and simplify the numerator. And now we have our answer:
Multiplication and Division of Rational Expressions
Multiplying Rational Expressions
Here are the three steps to multiplying rational expressions. Now, remember, when multiplying fractions, numerators and denominators are multiplied straight across.
- Step #1: Factor the numerator and denominator of each expression being multiplied.
- Step #2: Simplify by canceling out common factors from the numerator and the denominator.
- Step #3: The final answer is what is left after canceling. You may be asked to include domain restrictions with your solution.
Let’s use these steps to solve an example problem:
Now, because we have like terms in the numerator and the denominator position, we’re able to cancel them out. That leaves us with:
But we can simplify this even further, remember, 6 is a factor of 36, so let’s simplify:
And yet, we can simplify this again, remember, you have an
And now we have our answer,
Dividing Rational Expressions
Dividing rational expressions includes one extra step at the beginning of the process. When dividing by a fraction, it is the same as multiplying by the reciprocal of the second fraction. You can remember this rule as, “Keep, Change, and Flip” which translates to keep the first fraction, change the operation to multiplication, and take the reciprocal (or flip) of the second fraction.
Keep in mind that domain restrictions must be considered from both the numerator and denominator of the second fraction because of the “flip” in the division process.
Here’s an example:
Now, remember our three steps: keep the first fraction, change the operation, and then flip. Here we go:
Here is now where we multiply, cause we kept the first fraction, we changed to multiplication, and then we flipped the fraction over here. So, time to multiply.
So now we have our answer:
But remember, that’s not our complete answer if we don’t include our restricted domain, we have
I hope this review was helpful! See you next time!
Rational Expression Practice Questions
Which polynomial is considered a binomial?
Add the following polynomials:
What are the domain restrictions for the following expression?
Divide the following: