Intro to adding & subtracting rational expressions

CCSS Math: HSA.APR.D.7
Learn how to add or subtract two rational expressions into a single expression.

What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. For example, the expression x+2x+1\dfrac{x+2}{x+1} is a rational expression.
If you are unfamiliar with rational expressions, you may want to check out our intro to rational expressions.

What you will learn in this lesson

In this lesson, you will learn how to add and subtract rational expressions.

Adding and subtracting rational expressions (common denominators)

Numerical fractions

We can add and subtract rational expressions in much the same way as we add and subtract numerical fractions.
To add or subtract two numerical fractions with the same denominator, we simply add or subtract the numerators, and write the result over the common denominator.
=4515=415=35\begin{aligned} &\phantom{=}\dfrac{\blueD4}{\purpleC5}-\dfrac{\blueD1}{\purpleC5}\\\\\\ &=\dfrac{\blueD{4}-\blueD{1}}{\purpleC 5}\\ \\ &=\dfrac{3}{5} \end{aligned}

Variable expressions

The process is the same with rational expressions:
=7a+3a+2+2a1a+2=(7a+3)+(2a1)a+2Add=7a+3+2a1a+2Remove parentheses=9a+2a+2Combine like terms\begin{aligned} &\phantom{=}\dfrac{\blueD{7a+3}}{\purpleC{a+2}}+\dfrac{\blueD{2a-1}}{\purpleC{a+2}}\\\\\\ &=\dfrac{(\blueD{7a+3})+(\blueD{2a-1})}{\purpleC{a+2}}&&\small{\gray{\text{Add}}}\\ \\ &=\dfrac{{7a+3}+{2a-1}}{{a+2}}&&\small{\gray{\text{Remove parentheses}}}\\ \\ &=\dfrac{9a+2}{a+2}&&\small{\gray{\text{Combine like terms}}} \end{aligned}
It is good practice to place the numerators in parentheses, especially when subtracting rational expressions. This way, we are reminded to distribute the negative sign!
For example:
=b+1b24bb2=(b+1)(4b)b2Subtract=b+14+bb2Remove parentheses and distribute=2b3b2Combine like terms\begin{aligned} &\phantom{=}\dfrac{\blueD{b+1}}{\purpleC{b^2}}-\dfrac{\blueD{4-b}}{\purpleC{b^2}}\\\\\\ &=\dfrac{(\blueD{b+1})-(\blueD{4-b})}{\purpleC{b^2}}&&\small{\gray{\text{Subtract}}}\\ \\ &=\dfrac{b+1-4+b}{{b^2}}&&\small{\gray{\text{Remove parentheses and distribute}}}\\ \\ &=\dfrac{2b-3}{b^2}&&\small{\gray{\text{Combine like terms}}} \end{aligned}

Check your understanding

Adding and subtracting rational expressions (different denominators)

Numerical fractions

To understand how to add or subtract rational expressions with different denominators, let's first examine how this is done with numerical fractions.
For example, let's find 23+12\dfrac23+\dfrac12.
=23+12=23(22)+12(33)Create common denominators=46+36=76\begin{aligned} &\phantom{=}\dfrac{2}{\blueD3}+\dfrac{1}{\tealD2}\\\\\\ &=\dfrac{2}{\blueD3} \left(\tealD{\dfrac{2}{2}}\right)+\dfrac{1}{\tealD2}\left( \blueD{\dfrac{3}{3}}\right)&&\small{\gray{\text{Create common denominators}}}\\ \\ &=\dfrac{4}{6}+\dfrac{3}{6}\\ \\ &=\dfrac{7}{6} \end{aligned}
Notice that a common denominator of 66 was needed to add the two fractions:
  • The denominator in the first fraction (3)(\blueD 3) needed a factor of 2\tealD 2.
  • The denominator in the second fraction (2)(\tealD 2) needed a factor of 3\blueD3.
Each fraction was multiplied by a form of 11 to obtain this.

Variable expressions

Now let's apply this to the following example:
1x3+2x+5\dfrac{1}{\blueD{x-3}}+\dfrac{2}{\tealD{x+5}}
In order for the two denominators to be the same, the first needs a factor of x+5\tealD{x+5} and the second needs a factor of x3\blueD{x-3}. Let's manipulate the fractions in order to achieve this. Then, we can add as usual.
=1x3+2x+5=1x3(x+5x+5)+2x+5(x3x3)Create common denominators=1(x+5)(x3)(x+5)+2(x3)(x+5)(x3)=1(x+5)+2(x3)(x3)(x+5)Add=1x+5+2x6(x3)(x+5)=3x1(x3)(x+5)\begin{aligned} &\phantom{=}{\dfrac{1}{\blueD{x-3}}+\dfrac{2}{\tealD{x+5}}}\\\\\\ &=\dfrac{1}{\blueD{x-3}}{\left(\tealD{\dfrac{x+5}{x+5}}\right)}+\dfrac{2}{\tealD{x+5}}{\left(\blueD{\dfrac{x-3}{x-3}}\right)}&&\small{\gray{\text{Create common denominators}}}\\\\\\ &=\dfrac{1(x+5)}{(x-3)(x+5)}+\dfrac{2(x-3)}{(x+5)(x-3)}\\ \\\\\\ &=\dfrac{1(x+5)+2(x-3)}{(x-3)(x+5)}&&\small{\gray{\text{Add}}}\\ \\\\\\ &=\dfrac{1x+5+2x-6}{(x-3)(x+5)}\\ \\\\\\ &=\dfrac{3x-1}{(x-3)(x+5)} \end{aligned}
Notice that the first step is possible because x+5x+5\dfrac{x+5}{x+5} and x3x3\dfrac{x-3}{x-3} are equal to 11, and multiplication by 11 does not change the value of the expression!
In the last two steps, we simplified the numerator. While you can also multiply (x3)(x-3) and (x+5)(x+5) in the denominator, it is common to leave this in factored form.

Check your understanding

What's next?

Our next article covers more challenging examples of adding and subtracting rational expressions.
You will learn about the least common denominator, and why it is important to use this as the common denominator when adding or subtracting rational expressions.
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