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Rational Expressions

Rational Expressions Video

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 TermsType
4, 5x, 3x4, 8xy21Monomial
(x+3), (x21), (3xy+2y)2Binomial
(x2+5x+6),
(x22xy+y2)
3Trinomial

 

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 p(x) and g(x), a rational expression can be defined like this:

p(x)g(x) where g(x)0

 
Here’s an example:

5xx2

 
This example shows a rational expression with a monomial, 5x, in the numerator and a binomial, (x2), in the denominator. The value x=2 is the excluded value, as it would result in a denominator of 0. This expression cannot be simplified further.

5xx2, x2

 

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:

15+37

 
In order to add 15+37, we must create a common denominator. Specifically, we need to determine the least common denominator, meaning the smallest multiple of 5 and 7. In this case, that number is 35. The adjustment to each fraction that needs to be made to create the common denominator is:

15(77)+37(55)
735+1535
=2235

 
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 55, in order to create 35 in the denominator. After these adjustments are made and the denominators are the same, simplify the numerators:

7+15352235

 
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:

3xx2+5x2x2
3xx2+5(x2)(x+1)

 
Now we want to determine the lowest common denominator. What is the smallest multiple of (x2) and (x2)(x+1)?

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 x2, or over here, -1. If x=1 this would end up being 0, multiplied by another term, still remains 0. The 0 in the denominator, we can’t have that. Over here, if x=2, 22=0, again, we can’t have a 0 in the denominator, so these are our two terms, our domains that need to be excluded.

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 x+1 in the numerator and the denominator.

3xx2(x+1)(x+1)+5(x2)(x+1)

 
Now we’re going to rewrite the expression as a fraction, and simplify the numerator. And now we have our answer:

3x2+3x(x+1)(x2)+5(x2)(x+1)=3x2+3x+5(x2)(x+1)

 

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.

  1. Step #1: Factor the numerator and denominator of each expression being multiplied.
  2. Step #2: Simplify by canceling out common factors from the numerator and the denominator.
  3. 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:

3x4x82x24x9x
3x4(x2)2x(x2)9x
6x2(x2)36x(x2)

 
Now, because we have like terms in the numerator and the denominator position, we’re able to cancel them out. That leaves us with:

6x236x

 

But we can simplify this even further, remember, 6 is a factor of 36, so let’s simplify:

x26x

 
And yet, we can simplify this again, remember, you have an x in the numerator and an x in the denominator, so let’s simplify:

x6

 
And now we have our answer, x6. But that’s not the complete answer. Remember, we have some domain that we have to exclude. Up here, x2 because 2×48=0. And we can’t have a 0 in the denominator. So 2 is out, x2. Also, x0, because 0×9=0, and again, give us a 0 in the denominator. So the domains we have to exclude from this answer are 2 and 0. So our answer is x6,x2,0.

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:

9x2x2+12x+36÷12xx2+6x

 
Now, remember our three steps: keep the first fraction, change the operation, and then flip. Here we go:

9x2x2+12x+36×x2+6x12x, x0, 6

 
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.

9x2(x+6)(x+6)(x+6)12=3x24(x+6)

 
So now we have our answer: 3x24(x+6).

But remember, that’s not our complete answer if we don’t include our restricted domain, we have x0, and x6. Remember we have to make sure that we don’t have a 0 in the denominator or the numerator of our second term.

I hope this review was helpful! See you next time!

Rational Expression Practice Questions

Question #1:

 
Which polynomial is considered a binomial?

2x
x2+2x4
100
3x7
Question #2:

 
Add the following polynomials: x+23x+x36x

3x+1
3x+16x
6x+13x
6x
Question #3:

 
What are the domain restrictions for the following expression?
7x+2x24

x2 and x2
x2
x2
x7 and x4
Question #4:

 
12x25y3×20y46x3

8yx2
6yx
8yy2
8yx
Question #5:

 
Divide the following: x2x123x15÷x2924x72

8(x4)x5x5,3,3
x3x5x5
8(x+4)x+5x3
x+4x5x5,4,6
415183

 

by Mometrix Test Preparation | Last Updated: February 6, 2025