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Adding and Subtracting Rational Expressions

Author: Sophia

what's covered
In this lesson, you will learn how to add or subtract two rational expressions with an uncommon denominator. Specifically, this lesson will cover:

Table of Contents

1. Adding and Subtracting Numeric Fractions

Reviewing how to add and subtract numeric fractions is helpful when learning how to add and subtract algebraic fractions. This is because the same general principle is applied in both cases. In order to add two fractions, we want to express each fraction such that the denominators are the same. Then we can simply add or subtract across the numerators and retain the common denominator.

EXAMPLE

When adding 5 over 7 plus 3 over 7, this is fairly straightforward. The denominators are already equivalent, so we can just add the numerators 5 and 3 to get a sum of 8 over 7.

When the denominators are not the same, we need to write equivalent equations with the same denominator.

EXAMPLE

Consider the subtraction problem 5 over 6 minus 1 half.

5 over 6 minus 1 half Multiply 1 half by 3 over 3
5 over 6 plus 1 half times 3 over 3 Evaluate multiplication
5 over 6 minus 3 over 6 Subtract numerators
fraction numerator 5 minus 3 over denominator 6 end fraction Evaluate numerator
2 over 6 Simplify
1 third Our solution


2. Adding Rational Expression with Common Denominators

When the denominators between two factions are the same, adding and subtracting is much easier. This is also the case with rational expressions. Below is an example of adding rational expressions with the same denominator:

EXAMPLE

Add fraction numerator 2 x plus 2 over denominator 4 x minus 4 end fraction plus fraction numerator 3 x minus 6 over denominator 4 x minus 4 end fraction.

Since the denominators are the same, we can simply add the numerators together and keep the denominator.

fraction numerator 2 x plus 2 over denominator 4 x minus 4 end fraction plus fraction numerator 3 x minus 6 over denominator 4 x minus 4 end fraction Add numerators, keep denominator
fraction numerator open parentheses 2 x plus 2 close parentheses plus open parentheses 3 x minus 6 close parentheses over denominator 4 x minus 4 end fraction Evaluate numerator
fraction numerator 5 x minus 4 over denominator 4 x minus 4 end fraction Our solution

big idea
Adding and subtracting rational expressions with common denominators operates in the same way as adding and subtracting numeric fractions with common denominators. We add or subtract across the numerators, and keep the denominator the same.


3. Adding and Subtracting Rational Expressions with Uncommon Denominators

When adding or subtracting rational expressions without a common denominator, we must re-express the fractions so that they do have a common denominator. That way, we can add or subtract as we did in the example above.

The easiest way to find a common denominator between two algebraic fractions is to follow these steps:

  1. Multiply both the numerator and the denominator of the first fraction by the denominator of the second fraction.
  2. Repeat the process with the second fraction: multiply both the numerator and the denominator of the second fraction by the denominator of the first fraction.
  3. Add or subtract numerators and keep the denominator the same.

EXAMPLE

Subtact fraction numerator x plus 2 over denominator x end fraction minus fraction numerator 7 over denominator x minus 3 end fraction.

Our first task is to find a common denominator. To do this, multiply the numerator and the denominator of the first fraction fraction numerator x plus 2 over denominator x end fraction by a fraction with the denominator of the second fraction, x minus 3, in both the numerator and denominator, fraction numerator x minus 3 over denominator x minus 3 end fraction.

fraction numerator x plus 2 over denominator x end fraction Multiply by a fraction made from the denominator of the second fraction, fraction numerator x minus 3 over denominator x minus 3 end fraction
fraction numerator x plus 2 over denominator x end fraction open parentheses fraction numerator x minus 3 over denominator x minus 3 end fraction close parentheses Multiply numerators together by FOILing; denominator is product of the two denominators
fraction numerator x squared minus 3 x plus 2 x minus 6 over denominator x open parentheses x minus 3 close parentheses end fraction Combine like terms
fraction numerator x squared minus x minus 6 over denominator x open parentheses x minus 3 close parentheses end fraction Our new first fraction

We will repeat this process with the second fraction. Multiply the numerator and the denominator of the second fraction fraction numerator 7 over denominator x minus 3 end fraction by a fraction with the denominator of the first fraction, x, in both the numerator and denominator, x over x.

fraction numerator 7 over denominator x minus 3 end fraction Multiply by a fraction made from the denominator of the first fraction, x over x
fraction numerator 7 over denominator x minus 3 end fraction open parentheses x over x close parentheses Multiply numerators together; denominator is product of the two denominators
fraction numerator 7 x over denominator x open parentheses x minus 3 close parentheses end fraction Our new second fraction

Now that both fractions have the same denominator, we can subtract.

fraction numerator x plus 2 over denominator x end fraction minus fraction numerator 7 over denominator x minus 3 end fraction Replace fractions with results from above
fraction numerator x squared minus x minus 6 over denominator x open parentheses x minus 3 close parentheses end fraction minus fraction numerator 7 x over denominator x open parentheses x minus 3 close parentheses end fraction Subtract numerators; denominator stays the same
fraction numerator open parentheses x squared minus x minus 6 close parentheses minus open parentheses 7 x close parentheses over denominator x open parentheses x minus 3 close parentheses end fraction Evaluate numerator
fraction numerator x squared minus x minus 6 minus 7 x over denominator x open parentheses x minus 3 close parentheses end fraction Combine like terms
fraction numerator x squared minus 8 x minus 6 over denominator x open parentheses x minus 3 close parentheses end fraction Our solution

The expression fraction numerator x plus 2 over denominator x end fraction minus fraction numerator 7 over denominator x minus 3 end fraction can be written as fraction numerator x squared minus 8 x minus 6 over denominator x left parenthesis x minus 3 right parenthesis end fraction.

hint
This can be the easier strategy, however, this can get messy, because there may be common factors in the numerator and denominator that should be canceled, but not necessarily easy to spot.

EXAMPLE

Add fraction numerator x plus 1 over denominator 5 x end fraction plus fraction numerator 2 x plus 3 over denominator 3 x squared end fraction.

fraction numerator x plus 1 over denominator 5 x end fraction plus fraction numerator 2 x plus 3 over denominator 3 x squared end fraction Multiply each fraction by a fraction made up with the denominator of the other
fraction numerator x plus 1 over denominator 5 x end fraction open parentheses fraction numerator 3 x squared over denominator 3 x squared end fraction close parentheses plus fraction numerator 2 x plus 3 over denominator 3 x squared end fraction open parentheses fraction numerator 5 x over denominator 5 x end fraction close parentheses Evaluate the multiplication in each numerator
fraction numerator 3 x cubed plus 3 x squared over denominator 5 x open parentheses 3 x squared close parentheses end fraction plus fraction numerator 10 x squared plus 15 x over denominator 5 x open parentheses 3 x squared close parentheses end fraction Add numerators; denominator stays the same
fraction numerator open parentheses 3 x cubed plus 3 x squared close parentheses plus open parentheses 10 x squared plus 15 x close parentheses over denominator 5 x open parentheses 3 x squared close parentheses end fraction Combine like terms
fraction numerator 3 x cubed plus 13 x squared plus 15 x over denominator 5 x open parentheses 3 x squared close parentheses end fraction Simplify numerator by factoring out common term, x
fraction numerator x open parentheses 3 x squared plus 13 x plus 15 close parentheses over denominator 5 x open parentheses 3 x squared close parentheses end fraction Cancel x from both numerator and denominator
fraction numerator 3 x squared plus 13 x plus 15 over denominator 5 open parentheses 3 x squared close parentheses end fraction Our solution

The expression fraction numerator x plus 1 over denominator 5 x end fraction plus fraction numerator 2 x plus 3 over denominator 3 x squared end fraction can be written as fraction numerator 3 x squared plus 13 x plus 15 over denominator 5 left parenthesis 3 x squared right parenthesis end fraction.

summary
Recall when adding and subtracting numeric fractions, the fractions must have a common denominator. To add or subtract rational expressions, write each fraction with a common denominator then add or subtract the numerators, keeping the denominator the same. To find a common denominator, multiply both the numerator and the denominator of the first fraction by the denominator of the second fraction. Repeat this process with the second fraction.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License