Adding & subtracting rational expressions (advanced)

CCSS Math: HSA.APR.D.7
Have you learned the basics of rational expression addition/subtraction? Great! Now dig deeper with some advanced examples.

What we need to know before this lesson

A rational expression is a ratio of two polynomials.
To add or subtract two rational expressions with the same denominator, we simply add or subtract the numerators and write the result over the common denominator.
When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator.
If this is new to you, you may want to check out the following articles first:

What you will learn in this lesson

In this lesson, you will practice adding and subtracting rational expressions with different denominators. You will use the least common denominator as your common denominator in these examples and explore why it is beneficial to do so.

Warm-up: 3x22x+1\dfrac{3}{x-2}-\dfrac{2}{x+1}

To subtract two rational expression, each fraction must have the same denominator!
In this example, we can create a common denominator by multiplying the first fraction by (x+1x+1)\left({\dfrac{x+1}{x+1}}\right) and the second fraction by (x2x2)\left({\dfrac{x-2}{x-2}}\right).
In order for 3x22x+1\dfrac{3}{x-2}-\dfrac{2}{x+1} to be defined, x2,1x\neq 2, -1.
Therefore, x+1x+1\dfrac{x+1}{x+1} and x2x2\dfrac{x-2}{x-2} are both equal to 11, and multiplication by 11 does not change the value of the expression!
Then, we can subtract the numerators and write the result over the common denominator.
=3x22x+1=3x2(x+1x+1)2x+1(x2x2)Common denominator=3(x+1)(x2)(x+1)2(x2)(x+1)(x2)=3(x+1)2(x2)(x2)(x+1)Subtract=3x+32x+4(x2)(x+1)=x+7(x2)(x+1)\begin{aligned}&\phantom{=}{\dfrac{3}{\blueD{x-2}}-\dfrac{2}{\greenD{x+1}}}\\\\\\ &=\dfrac{3}{\blueD{x-2}}{\left(\greenD{\dfrac{x+1}{x+1}}\right)}-\dfrac{2}{\greenD{x+1}}{\left(\blueD{\dfrac{x-2}{x-2}}\right)}&&\small{\gray{\text{Common denominator}}}\\\\\\ &=\dfrac{3(x+1)}{(x-2)(x+1)}-\dfrac{2(x-2)}{(x+1)(x-2)}\\ \\\\\\ &=\dfrac{3(x+1)-2(x-2)}{(x-2)(x+1)}&&\small{\gray{\text{Subtract}}}\\ \\\\\\ &=\dfrac{3x+3-2x+4}{(x-2)(x+1)}\\ \\\\\\ &=\dfrac{x+7}{(x-2)(x+1)} \end{aligned}

Check your understanding

1) 5xx+3+4x+2=\dfrac{5x}{x+3}+\dfrac{4}{x+2}=
=5xx+3+4x+2=5xx+3(x+2x+2)+4x+2(x+3x+3)Common denominator=5x(x+2)(x+3)(x+2)+4(x+3)(x+3)(x+2)=5x(x+2)+4(x+3)(x+3)(x+2)Add=5x2+10x+4x+12(x+3)(x+2)=5x2+14x+12(x+3)(x+2)\begin{aligned}&\phantom{=}\dfrac{5x}{\blueD{x+3}}+\dfrac{4}{\greenD{x+2}}\\\\\\ &=\dfrac{5x}{\blueD{x+3}}\left(\greenD{\dfrac{x+2}{x+2}}\right)+\dfrac{4}{\greenD{x+2}}\left(\blueD{\dfrac{x+3}{x+3}}\right)&&\small{\gray{\text{Common denominator}}}\\ \\\\\\ &=\dfrac{5x(x+2)}{(x+3)(x+2)}+\dfrac{4(x+3)}{(x+3)(x+2)}\\ \\\\\\ &=\dfrac{5x(x+2)+4(x+3)}{(x+3)(x+2)}&&\small{\gray{\text{Add}}}\\ \\\\\\ &=\dfrac{5x^2+10x+4x+12}{(x+3)(x+2)}\\ \\\\\\ &=\dfrac{5x^2+14x+12}{(x+3)(x+2)} \end{aligned}

Least common denominators

Numerical fractions

Sometimes, the denominators of two fractions are different but have some shared factors.
For example, consider 34+16\dfrac{3}{4}+\dfrac{1}{6}:
=34+16=322+123=322(33)+123(22)Create least common denominator=912+212=1112\begin{aligned} &\phantom{=}\dfrac{3}{4}+\dfrac{1}{6}\\\\\\ &=\dfrac{3}{\blueD2\cdot \greenD2}+\dfrac{1}{\blueD2\cdot \goldD3}\\ \\ &=\dfrac{3}{\blueD2\cdot \greenD2} {\left(\dfrac{\goldD3}{\goldD3}\right)}+\dfrac{1}{\blueD2\cdot\goldD3}{\left(\dfrac{\greenD2}{\greenD2}\right)}&&\small{\gray{\text{Create least common denominator}}}\\ \\ &=\dfrac{9}{12}+\dfrac{2}{12}\\ \\ &=\dfrac{11}{12} \end{aligned}
Notice that the common denominator used in this example was not the product of the two individual denominators (24)(24). Instead it was the least common multiple of 44 and 66 (12)(12).
The least common multiple (LCM)(\text{LCM}) of two integers is the smallest number that is divisible by both integers.
When two integers do not share a common factor, the LCM\text{LCM} is simply equal to the product of these integers. However, when two integers share a common factor, the LCM\text{LCM} is not equal to the product of the integers. This is because the common factor appears only once in the LCM\text{LCM}. For instance:
  • 44 and 66 share a common factor of 22, so the LCM\text{LCM} of 44 and 66 is 1212, not 46=244\cdot 6=24
  • 66 and 1515 share a common factor of 33, so the LCM\text{LCM} of 66 and 1515 is 3030, not 615=906\cdot 15=90
The least common multiple of the denominators in two or more fractions is called the least common denominator.
An easy way to find the least common denominator is to think about how to make the denominators equal without having any unnecessary factors. Writing each denominator in its prime factorization makes this easy to see.
Notice that 223=12\blueD2\cdot \greenD2\cdot \goldD3=12

Variable expressions

Now let's apply this reasoning to perform the following addition:
2(x2)(x+1)+3(x+1)(x+3)\dfrac{2}{(x-2)(x+1)}+\dfrac{3}{(x+1)(x+3)}
First, let's find the least common denominator:
So the least common denominator is (x2)(x+1)(x+3)\blueD{(x-2)}\greenD{(x+1)}\purpleC{(x+3)}.
We can add the rational expressions as follows:
=2(x2)(x+1)+3(x+1)(x+3)=2(x2)(x+1)(x+3x+3)+3(x+1)(x+3)(x2x2)Least common denominator=2(x+3)(x2)(x+1)(x+3)+3(x2)(x+1)(x+3)(x2)=2(x+3)+3(x2)(x2)(x+1)(x+3)Add=2x+6+3x6(x2)(x+1)(x+3)=5x(x2)(x+1)(x+3)\small\begin{aligned}&\phantom{=}\dfrac{2}{\blueD{(x-2)}\greenD{(x+1)}}+\dfrac{3}{\greenD{(x+1)}\purpleC{(x+3)}}\\ \\\\ &=\dfrac{2}{\blueD{(x-2)}\greenD{(x+1)}}{\left(\purpleC{\dfrac{x+3}{x+3}}\right)}+\dfrac{3}{\greenD{(x+1)}\purpleC{(x+3)}}{\left(\blueD{\dfrac{x-2}{x-2}}\right)}&&\tiny{\gray{\text{Least common denominator}}}\\\\ \\\\ &=\dfrac{2(x+3)}{(x-2)(x+1)(x+3)}+\dfrac{3(x-2)}{(x+1)(x+3)(x-2)}\\ \\\\ &=\dfrac{2(x+3)+3(x-2)}{(x-2)(x+1)(x+3)}&&\small{\gray{\text{Add}}}\\ \\\\ &=\dfrac{2x+6+3x-6}{(x-2)(x+1)(x+3)}\\ \\\\ &=\dfrac{5x}{(x-2)(x+1)(x+3)} \end{aligned}

Check your understanding

2) 1x(x6)+3(x+1)(x6)=\dfrac{1}{x(x-6)}+\dfrac{3}{(x+1)(x-6)}=
First, let's find the least common denominator:
So the least common denominator is x(x6)(x+1)\blueD{x}\greenD{(x-6)}\purpleC{(x+1)}.
We can add the rational expressions as follows:
=1x(x6)+3(x+1)(x6)=1x(x6)(x+1x+1)+3(x+1)(x6)(xx)Least common denominator=1(x+1)x(x6)(x+1)+3xx(x6)(x+1)=1(x+1)+3xx(x6)(x+1)Add=x+1+3xx(x6)(x+1)=4x+1x(x6)(x+1)\small\begin{aligned}&\phantom{=}\dfrac{1}{\blueD x\greenD{(x-6)}}+\dfrac{3}{\purpleC{(x+1)}\greenD{(x-6)}}\\\\\\ &=\dfrac{1}{\blueD x\greenD{(x-6)}}\left(\purpleC{\dfrac{x+1}{x+1}}\right)+\dfrac{3}{\purpleC{(x+1)}\greenD{(x-6)}}\left(\blueD{\dfrac{x}{x}}\right)&&\tiny{\gray{\text{Least common denominator}}}\\\\ \\ &=\dfrac{1(x+1)}{x(x-6)(x+1)}+\dfrac{3x}{x(x-6)(x+1)}\\ \\\\ &=\dfrac{1(x+1)+3x}{x(x-6)(x+1)}&&\small{\gray{\text{Add}}}\\ \\\\ &=\dfrac{x+1+3x}{x(x-6)(x+1)}\\ \\\\ &=\dfrac{4x+1}{x(x-6)(x+1)} \end{aligned}
3) 3x2(x1)4(x1)(x+2)=\dfrac{3x}{2(x-1)}-\dfrac{4}{(x-1)(x+2)}=
First, let's find the least common denominator:
So the least common denominator is 2(x1)(x+2)\blueD{2}\greenD{(x-1)}\purpleC{(x+2)}.
We can subtract the rational expressions as follows:
=3x2(x1)4(x1)(x+2)=3x2(x1)(x+2x+2)4(x1)(x+2)(22)Least common denominator=3x(x+2)2(x1)(x+2)82(x1)(x+2)=3x(x+2)82(x1)(x+2)Subtract=3x2+6x82(x1)(x+2)\small\begin{aligned}&\phantom{=}\dfrac{3x}{\blueD 2\greenD{(x-1)}}-\dfrac{4}{\greenD{(x-1)}\purpleC{(x+2)}}\\\\\\ &=\dfrac{3x}{\blueD 2\greenD{(x-1)}}\left(\purpleC{\dfrac{x+2}{x+2}}\right)-\dfrac{4}{\greenD{(x-1)}\purpleC{(x+2)}}\left(\blueD{\dfrac{2}{2}}\right)&&\tiny{\gray{\text{Least common denominator}}}\\ \\\\ &=\dfrac{3x(x+2)}{2(x-1)(x+2)}-\dfrac{8}{2(x-1)(x+2)}\\ \\\\ &=\dfrac{3x(x+2)-8}{2(x-1)(x+2)}&&\small{\gray{\text{Subtract}}}\\ \\ &=\dfrac{3x^2+6x-8}{2(x-1)(x+2)} \end{aligned}

Challenge Problem

4*) 2x21+1x23x4=\dfrac{2}{x^2-1}+\dfrac{1}{x^2-3x-4}=
Notice that in this case, the denominators are not in factored form. So let's factor first.
2x21+1x23x4=2(x1)(x+1)+1(x4)(x+1)\dfrac{2}{x^2-1}+\dfrac{1}{x^2-3x-4}=\dfrac{2}{(x-1)(x+1)}+\dfrac{1}{(x-4)(x+1)}
Now it is easier to find the least common denominator.
The least common denominator is (x1)(x+1)(x4)\blueD{(x-1)}\greenD{(x+1)}\purpleC{(x-4)}.
We can add the rational expressions as follows:
=2(x1)(x+1)+1(x4)(x+1)=2(x1)(x+1)(x4x4)+1(x4)(x+1)(x1x1)Least common denominator=2(x4)(x1)(x+1)(x4)+1(x1)(x4)(x+1)(x1)=2(x4)+1(x1)(x1)(x+1)(x4)Add=2x8+x1(x1)(x+1)(x4)=3x9(x1)(x+1)(x4)\small\begin{aligned}&\phantom{=}\dfrac{2}{\blueD {(x-1)}\greenD{(x+1)}}+\dfrac{1}{\purpleC{(x-4)}\greenD{(x+1)}}\\\\\\ &=\dfrac{2}{\blueD {(x-1)}\greenD{(x+1)}}\left(\purpleC{\dfrac{x-4}{x-4}}\right)+\dfrac{1}{\purpleC{(x-4)}\greenD{(x+1)}}\left(\blueD{\dfrac{x-1}{x-1}}\right)&&\tiny{\gray{\text{Least common denominator}}}\\ \\\\ &=\dfrac{2(x-4)}{(x-1)(x+1)(x-4)}+\dfrac{1(x-1)}{(x-4)(x+1)(x-1)}\\ \\\\ &=\dfrac{2(x-4)+1(x-1)}{(x-1)(x+1)(x-4)}&&\small{\gray{\text{Add}}}\\ \\\\ &=\dfrac{2x-8+x-1}{(x-1)(x+1)(x-4)}\\\\\\ &=\dfrac{3x-9}{(x-1)(x+1)(x-4)} \end{aligned}

Why use the least common denominator?

You may be wondering why it is so important to use the least common denominator to add or subtract rational expressions.
After all, this is not a requirement, and it is easy enough to use other denominators with numerical fractions.
For example, the table below calculates 34+16\dfrac{3}{4}+\dfrac{1}{6} using two different common denominators — one using the least common denominator (12)(12) and the other using the product of the two denominators (24)(24).
Least common denominator (12)(12)     ~~~~~Common denominator (24)(24)
 34+16=34(33)+16(22)=912+212=111212\begin{aligned}~\dfrac{3}{4}+\dfrac{1}{6}&=\dfrac{3}{\blueD4} {\left(\dfrac{\goldD3}{\goldD3}\right)}+\dfrac{1}{\greenD6}{\left(\dfrac{\purpleD2}{\purpleD2}\right)}\\\\&=\dfrac{9}{12}+\dfrac{2}{12}\\\\&=\dfrac{11}{12}\\\\&\phantom{\dfrac{1}{2}}\end{aligned}       34+16=34(66)+16(44)=1824+424=2224=1112~~~~~~~\begin{aligned}\dfrac{3}{\blueD4}+\dfrac{1}{\greenD6}&=\dfrac{3}{\blueD4}\left(\greenD{\dfrac{6}{6}}\right)+\dfrac{1}{\greenD6}\left(\blueD{\dfrac{4}{4}}\right)\\\\&=\dfrac{18}{24}+\dfrac{4}{24}\\\\&=\dfrac{22}{24}\\\\&=\dfrac{11}{12}\end{aligned}

Notice that when using 2424 as the common denominator, more work was required. The numbers were larger and the resulting fraction needed to be simplified.
This will also happen if you do not use the least common denominator when adding or subtracting rational expressions.
However, with rational expressions, this process is much more difficult because the numerators and denominators will be polynomials instead of integers! You will have to perform arithmetic with higher degree polynomials and factor polynomials to simplify the fraction.
Recall the following problem.
2(x2)(x+1)+3(x+1)(x+3)\dfrac{2}{(x-2)(x+1)}+\dfrac{3}{(x+1)(x+3)}
Suppose we obtained a common denominator by multiplying the first fraction by the denominator of the second and the second fraction by the denominator of the first. Note that this is a common denominator but not the least common denominator.
=2(x2)(x+1)+3(x+1)(x+3)=2(x2)(x+1)((x+1)(x+3)(x+1)(x+3))+3(x+1)(x+3)((x2)(x+1)(x2)(x+1))\scriptsize\begin{aligned}&\phantom{=}\dfrac{2}{\blueD{(x-2){(x+1)}}}+\dfrac{3}{\greenD{(x+1)(x+3)}}\\ \\\\ &=\dfrac{2}{\blueD{(x-2){(x+1)}}}{\left(\greenD{\dfrac{(x+1)(x+3)}{(x+1)(x+3)}}\right)}+\dfrac{3}{\greenD{(x+1)(x+3)}}{\left(\blueD{\dfrac{(x-2)(x+1)}{(x-2)(x+1)}}\right)} \end{aligned}
=2(x+1)(x+3)(x2)(x+1)2(x+3)+3(x2)(x+1)(x+1)2(x+3)(x2)=2(x+1)(x+3)+3(x2)(x+1)(x2)(x+1)2(x+3)Add=2(x2+4x+3)+3(x2x2)(x2)(x+1)2(x+3)=2x2+8x+6+3x23x6(x2)(x+1)2(x+3)=5x2+5x(x2)(x+1)2(x+3)=5x(x+1)(x2)(x+1)2(x+3)Factor out 5x=5x(x2)(x+1)(x+3)Cancel a factor of x+1\scriptsize\begin{aligned}&=\dfrac{2(x+1)(x+3)}{(x-2)(x+1)^2(x+3)}+\dfrac{3(x-2)(x+1)}{(x+1)^2(x+3)(x-2)}\\ \\\\\\ &=\dfrac{2(x+1)(x+3)+3(x-2)(x+1)}{(x-2)(x+1)^2(x+3)}&&\small{\gray{\text{Add}}}\\ \\\\ &=\dfrac{2(x^2+4x+3)+3(x^2-x-2)}{(x-2)(x+1)^2(x+3)}\\ \\\\ &=\dfrac{2x^2+8x+6+3x^2-3x-6}{(x-2)(x+1)^2(x+3)}\\ \\\\ &=\dfrac{5x^2+5x}{(x-2)(x+1)^2(x+3)}\\ \\\\ &=\dfrac{5x(x+1)}{(x-2)(x+1)^2(x+3)}&&\small{\gray{\text{Factor out }5x}}\\\\\\ &=\dfrac{5x}{(x-2)(x+1)(x+3)}&&\small{\gray{\text{Cancel a factor of }x+1}} \end{aligned}
Notice that while this gave you the correct answer in the end, much more work was required!
All of this extra work can be avoided by using the least common denominator when adding or subtracting rational expressions.
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