How to add and subtract rational expressions

 
 
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You always have to start with a common denominator

In this lesson we will look at how to add and subtract rational expressions. In other words, how to work with fractions that have variables in them as well as numbers.

Remember: when you add and subtract fractions you need a common denominator.

Let’s look at how to find the least common factor from a group of terms.

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Say you have, ???5x???, ???xy^2???, and ???5y^4???, you can set up a table to help you organize the factors. Put your terms along the left hand column and your parts along the top row. Then list the parts of each term.

 
table breaking down the factors
 

In order to generate the least common multiple, we have to take the largest value from each column of factors. The largest factor from the coefficients column is ???5???; the largest factor from the ???x???’s column is ???x???; the largest factor from the ???y???’s column is ???y^4???. Therefore, the least common multiple of our terms ???5x???, ???xy^2??? and ???5y^4??? is

???5xy^4???

Now let’s look at how to apply this idea to adding and subtracting rational expressions.

 
 

Four video examples of adding or subtracting rational expressions


 
 
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How to combine three fractions with different denominators

Example

Simplify the expression.

???\frac{y}{5x}+\frac{a}{xy^2}-\frac{c}{5y^4}???


We need to combine the three fractions in the expression into one fraction, which we’ll do by finding a common denominator.

The lowest common denominator will be the least common multiple of the three denominators in

???\frac{y}{5x}+\frac{a}{xy^2}-\frac{c}{5y^4}???

The denominators are, ???5x???, ???xy^2??? and ???5y^4???. We found that the common denominator is ???5xy^4???.

Now we need to multiply the numerator and denominator of each fraction by whatever value is required to make the denominator of the fraction ???5xy^4???.

???\frac{y^4}{y^4} \cdot \frac{y}{5x}+\frac{5y^2}{5y^2} \cdot \frac{a}{xy^2}-\frac{x}{x} \cdot \frac{c}{5y^4}???

???\frac{y^5}{5xy^4} +\frac{5ay^2}{5xy^4}- \frac{xc}{5xy^4}???

???\frac{y^5+5ay^2-xc}{5xy^4}???

adding and subtracting fractions

To find the common denominator, start by breaking each denominator down into its factors


Let’s do one more example.


Another example of how to simplify equations with parentheses


Example

Simplify the expression.

???\frac{a}{2bc^2}+\frac{m}{3c}+\frac{y}{4bc}???


We need to combine the three fractions in the expression into one fraction, which we’ll do by finding a common denominator.

The lowest common denominator will be the least common multiple of the three denominators in

???\frac{a}{2bc^2}+\frac{m}{3c}+\frac{y}{4bc}???

 
finding the common denominator to add the fractions
 

In order to generate the least common multiple, we have to take the largest common factor from each column of factors. The largest factor from the coefficients column is ???3 \cdot 2 \cdot 2 = 12???; the largest factor from the ???b???’s column is ???b???; the largest common factor from the ???c???’s column is ???c^2???. Therefore, the least common multiple is

???12bc^2???

Now we need to multiply the numerator and denominator of each fraction by whatever value is required to make the denominator of the fraction

???12bc^2???

We get

???\frac{6}{6} \cdot \frac{a}{2bc^2} + \frac{4bc}{4bc} \cdot \frac{m}{3c} + \frac{3c}{3c} \cdot \frac{y}{4bc}???

???\frac{6a}{12bc^2} +\frac{4bcm}{12bc^2} + \frac{3cy}{12bc^2}???

???\frac{6a+4bcm+3cy}{12bc^2}???

 
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