Learning Objectives
In this section students will:
In the previous section, we learned that a rational expression is the quotient of two polynomials. For example, if a pastry shop has fixed costs of [latex]\,\text{\$}280\,[/latex] per week and variable costs of [latex]\,\text{\$}9\,[/latex] per box of pastries, we saw that the average total cost per box of pastries is given by the rational expression
[latex]$$\frac{280+9x}{x}$$[/latex]
We’ve seen how to simplify, multiply, and divide rational expressions. In this section, we’ll learn how to add and subtract rational expressions.
6.2.1 – Adding and Subtracting Rational Expressions
Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.
[latex]$$\begin{array}{ccc}\hfill \frac{5}{24}+\frac{1}{40}& =& \frac{25}{120}+\frac{3}{120}\hfill \\ & =& \frac{28}{120}\hfill \\ & =& \frac{7}{30}\hfill \end{array}$$[/latex]
We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\,\left(x+3\right)\left(x+4\right)\,[/latex] and [latex]\,\left(x+4\right)\left(x+5\right),[/latex] then the LCD would be [latex]\,\left(x+3\right)\left(x+4\right)\left(x+5\right).[/latex]
Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\,\left(x+3\right)\left(x+4\right)\,[/latex] by [latex]\,\frac{x+5}{x+5}\,[/latex] and the expression with a denominator of [latex]\,\left(x+4\right)\left(x+5\right)\,[/latex] by [latex]\,\frac{x+3}{x+3}.[/latex]
How To
Given two rational expressions, add or subtract them.
- Factor the numerator and denominator.
- Find the LCD of the expressions.
- Multiply the expressions by a form of 1 that changes the denominators to the LCD.
- Add or subtract the numerators.
- Simplify.
Example 1 – Adding Rational Expressions
Add the rational expressions:
First, we have to find the LCD. In this case, the LCD will be [latex]\,xy.\,[/latex] We then multiply each expression by the appropriate form of 1 to obtain [latex]\,xy\,[/latex] as the denominator for each fraction.
[latex]$$\begin{array}{l}\frac{5}{x}\cdot \frac{y}{y}+\frac{6}{y}\cdot \frac{x}{x}\\ \frac{5y}{xy}+\frac{6x}{xy}\end{array}$$[/latex]
Now that the expressions have the same denominator, we simply add the numerators to find the sum.
[latex]$$\frac{6x+5y}{xy}$$[/latex]
Analysis
Multiplying by [latex]\,\frac{y}{y}\,[/latex] or [latex]\,\frac{x}{x}\,[/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.
Example 2 – Subtracting Rational Expressions
Subtract the rational expressions:
Do we have to use the LCD to add or subtract rational expressions?
No. Any common denominator will work, but it is easiest to use the LCD.
Try It
Subtract the rational expressions: [latex]\,\frac{3}{x+5}-\frac{1}{x-3}.[/latex]
Show answer
[latex]\frac{2\left(x-7\right)}{\left(x+5\right)\left(x-3\right)}[/latex]
6.2.2 – Simplifying Complex Rational Expressions
A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\,\frac{a}{\frac{1}{b}+c}\,[/latex] can be simplified by rewriting the numerator as the fraction [latex]\,\frac{a}{1}\,[/latex] and combining the expressions in the denominator as [latex]\,\frac{1+bc}{b}.\,[/latex] We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\,\frac{a}{1}\cdot \frac{b}{1+bc},[/latex] which is equal to [latex]\,\frac{ab}{1+bc}.[/latex]
How To
Given a complex rational expression, simplify it.
- Combine the expressions in the numerator into a single rational expression by adding or subtracting.
- Combine the expressions in the denominator into a single rational expression by adding or subtracting.
- Rewrite as the numerator divided by the denominator.
- Rewrite as multiplication.
- Multiply.
- Simplify.
Example 3 – Simplifying Complex Rational Expressions
Simplify: [latex]\frac{y+\frac{1}{x}}{\frac{x}{y}}[/latex] .
Begin by combining the expressions in the numerator into one expression.
[latex]$$\begin{array}{cc}y\cdot \frac{x}{x}+\frac{1}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Multiply by }\frac{x}{x}\text{ to get LCD as denominator}.\hfill \\ \frac{xy}{x}+\frac{1}{x}\hfill & \\ \frac{xy+1}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Add numerators}.\hfill \end{array}$$[/latex]
Now the numerator is a single rational expression and the denominator is a single rational expression.
[latex]$$\frac{\frac{xy+1}{x}}{\frac{x}{y}}$$[/latex]
We can rewrite this as division, and then multiplication.
[latex]$$\begin{array}{cc}\frac{xy+1}{x}÷\frac{x}{y}\hfill & \\ \frac{xy+1}{x}\cdot \frac{y}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{Rewrite as multiplication}\text{.}\hfill \\ \frac{y\left(xy+1\right)}{{x}^{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{Multiply}\text{.}\hfill \end{array}$$[/latex]
Try It
Simplify: [latex]\frac{\frac{x}{y}-\frac{y}{x}}{y}[/latex]
Show answer
[latex]\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}[/latex]
Can a complex rational expression always be simplified?
Yes. We can always rewrite a complex rational expression as a simplified rational expression.
Access these online resources for additional instruction and practice with rational expressions.
Key Concepts
Glossary
- least common denominator
- the smallest multiple that two denominators have in common
- rational expression
- the quotient of two polynomial expressions