Section 7.2Add and Subtract Rational Expressions

Definition

Before you get started, take this readiness quiz.

Add: $\frac{7}{10}+\frac{8}{15}.$
If you missed this problem, review Example 6.

$\frac{37}{30}$

Definition

Subtract: $\frac{3x}{4}-\frac{8}{9}.$
If you missed this problem, review Example 5.

$\frac{27x-32}{36}$

Definition

Subtract: $6(2x+1)-4(x-5).$
If you missed this problem, review Example 12.

$8x+26$

Add and Subtract Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

Rational Expression Addition and Subtraction

If p, q, and r are polynomials where $r\ne 0,$ then

\[\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\,\text{and}\,\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}\]

To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?

Example 1

Add: $\frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4}.$

The denominators already match, so add the numerators and place the sum over the common denominator.

Since the denominator is $x+4,$ we must exclude the value $x=-4.$

Table 1
	$\,\frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4},\,x\ne \text{-}4$
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	$\,\frac{11x+28+{x}^{2}}{x+4}$
Write the degrees in descending order.	$\,\frac{{x}^{2}+11x+28}{x+4}$
Factor the numerator.	$\,\frac{(x+4)(x+7)}{x+4}$
Simplify by removing common factors.	$\,\frac{(x+4)(x+7)}{x+4}$
Simplify.	$\,x+7$

The expression simplifies to $x+7$ but the original expression had a denominator of $x+4$ so $x\ne \text{-}4.$

Try It #1

Simplify: $\frac{9x+14}{x+7}+\frac{{x}^{2}}{x+7}.$

$x+2$

Did you get it?

Try It #2

Simplify: $\frac{{x}^{2}+8x}{x+5}+\frac{15}{x+5}.$

$x+3$

Did you get it?

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator. Be careful of the signs when you subtract a binomial or trinomial.

Example 2

Subtract: $\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}.$

Subtract the numerators, distributing the negative sign across the second numerator, then place the result over the common denominator.

Table 2
	$\,\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}$
Subtract the numerators and place the difference over the common denominator.	$\,\frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}$
Distribute the sign in the numerator.	$\,\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}$
Combine like terms.	$\,\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}$
Factor the numerator and the denominator.	$\,\frac{(x-2)(x-6)}{(x+3)(x-6)}$
Simplify by removing common factors.	$\,\frac{(x-2)(x-6)}{(x+3)(x-6)}$
	$\,\frac{(x-2)}{(x+3)}$

Try It #3

Subtract: $\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.$

$\frac{x-11}{x-2}$

Did you get it?

Try It #4

Subtract: $\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.$

$\frac{x-3}{x+9}$

Did you get it?

Add and Subtract Rational Expressions Whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by $\frac{-1}{-1}.$

Let’s see how this works.

Table 3
	$A mathematical expression showing the sum of two fractions: 7 over d plus 5 over negative d.$
Multiply the second fraction by $\frac{-1}{-1}.$	$A mathematical expression showing the sum of two fractions. The first fraction is 7 divided by d. The second fraction has a numerator of (-1) multiplied by 5, and a denominator of (-1) multiplied by (-d).$
The denominators are the same.	$A mathematical expression showing the addition of two fractions with a common denominator 'd'. The expression is 7/d + (-5)/d.$
Simplify.	$A mathematical fraction displaying the number 2 over the letter d, indicating 2 divided by d, centered on a white background.$

Be careful with the signs as you work with the opposites when the fractions are being subtracted.

Example 3

Subtract: $\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.$

Notice the denominators are opposites — multiply the second fraction by -1/-1 so both denominators match.

Table 4
	$The image displays a mathematical expression showing the subtraction of two algebraic fractions: (m^2 - 6m) / (m^2 - 1) - (3m + 2) / (1 - m^2).$
The denominators are opposites, so multiply the second fraction by $\frac{-1}{-1}.$	$An algebraic expression showing the subtraction of two rational terms: (m^2 - 6m)/(m^2 - 1) minus [-1(3m+2)]/[-1(1-m^2)], illustrating fraction manipulation.$
Simplify the second fraction.	$A mathematical expression displaying the subtraction of two algebraic fractions: (m^2 - 6m) / (m^2 - 1) - (-3m - 2) / (m^2 - 1). Both fractions share the same denominator, m^2 - 1.$
The denominators are the same. Subtract the numerators.	$A mathematical expression showing a fraction with m^2 - 6m - (-3m - 2) in the numerator and m^2 - 1 in the denominator.$
Distribute.	$A mathematical expression displaying the fraction (m^2 - 6m + 3m + 2) / (m^2 - 1), which simplifies to (m^2 - 3m + 2) / (m^2 - 1).$
Combine like terms.	$A mathematical expression displays a fraction with m squared minus 3m plus 2 in the numerator, and m squared minus 1 in the denominator.$
Factor the numerator and denominator.	$A mathematical fraction, where the numerator is (m-1)(m-2) and the denominator is (m-1)(m+1).$
Simplify by removing common factors.	$Cancellation of the common factor (m-1) from the numerator and denominator of the algebraic fraction ((m-1)(m-2))/((m-1)(m+1)).$
Simplify.	$A mathematical fraction is displayed, with 'm - 2' in the numerator and 'm + 1' in the denominator, set against a plain white background.$

Try It #5

Subtract: $\frac{{y}^{2}-5y}{{y}^{2}-4}-\frac{6y-6}{4-{y}^{2}}.$

$\frac{y+3}{y+2}$

Did you get it?

Try It #6

Subtract: $\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}.$

$\frac{3n-2}{n-1}$

Did you get it?

Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at this example: $\frac{7}{12}+\frac{5}{18}.$ Since the denominators are not the same, the first step was to find the least common denominator (LCD).

To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.

$Seven-twelfths plus five-eighteenths. Write the prime factorizations of each denominator and line up the common factors. The denominator of the first fraction is 12. The prime factorization of 12 is 2 times 2 times 3. The denominator of the second fraction is 18. The prime factorization of 18 is 2 times 3 times 3. Bringing down a factor from each column, the lowest common denominator of 12 and 18 is 2 times 2 times 3 times 3, which is 36. Write both fractions using the lowest common denominator. To do this multiply the numerator and denominator of the first fraction by 3 and multiply the numerator and denominator of the second fraction by 2. The result is 7 times 3 all divided by 12 times 3 plus 5 times 2 all divided by 18 times 2. Simplify each fraction. 7 times 3 is 21 and 12 times 3 is 36. 5 times 2 is 10 and 18 times 2 is 36. The result is twenty-one thirty-sixths plus ten thirty-sixths.$

We do the same thing for rational expressions. However, we leave the LCD in factored form.

Find the least common denominator of rational expressions.

Factor each denominator completely.
List the factors of each denominator. Match factors vertically when possible.
Bring down the columns by including all factors, but do not include common factors twice.
Write the LCD as the product of the factors.

Remember, we always exclude values that would make the denominator zero. What values of $x$ should we exclude in this next example?

Example 4

ⓐ Find the LCD for the expressions $\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}$ and ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

Factor each denominator completely, then line up the common factors to build the LCD.

ⓐ

Table 5
Find the LCD for $\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.$
Factor each denominator completely, lining up common factors. Bring down the columns.
Write the LCD as the product of the factors.

ⓑ

Table 6
	$Two algebraic fractions are displayed side-by-side. The first fraction is 8 over (x^2 - 2x - 3), followed by a prime symbol. The second fraction is 3x over (x^2 + 4x + 3).$
Factor each denominator.	$The image shows two algebraic fractions separated by a comma. The first fraction is 8 over (x+1)(x-3). The second fraction is 3x over (x+1)(x+3).$
Multiply each denominator by the ‘missing’ LCD factor and multiply each numerator by the same factor.	$Two algebraic fractions showing common factors highlighted in red, indicating terms like (x+3) and (x-3) that can be cancelled for simplification.$
Simplify the numerators.	$Two algebraic fractions are presented. The first is (8x + 24) / ((x + 1)(x - 3)(x + 3)) and the second is (3x^2 - 9x) / ((x + 1)(x + 3)(x - 3)).$

Try It #7

ⓐ Find the LCD for the expressions $\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}$ ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

ⓐ $(x-4)(x+3)(x+4)$
ⓑ $\frac{2x+8}{(x-4)(x+3)(x+4)}$ ,
$\frac{x+3}{(x-4)(x+3)(x+4)}$

Did you get it?

Try It #8

ⓐ Find the LCD for the expressions $\frac{3x}{{x}^{2}-3x-10},\frac{5}{{x}^{2}+3x+2}$ ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

ⓐ $(x+2)(x-5)(x+1)$
ⓑ $\frac{3{x}^{2}+3x}{(x+2)(x-5)(x+1)}$ ,
$\frac{5x-25}{(x+2)(x-5)(x+1)}$

Did you get it?

Add and Subtract Rational Expressions with Unlike Denominators

Now we have all the steps we need to add or subtract rational expressions with unlike denominators.

Example 5How to Add Rational Expressions with Unlike Denominators

Add: $\frac{3}{x-3}+\frac{2}{x-2}.$

Find the LCD of the two denominators, then rewrite each fraction over that LCD before adding.

Step 1 is to determine if the rational expressions 3 divided by the quantity x minus 3 and 2 divided by the quantity x minus 2 have a common factors. The denominators x minus 3 and x minus 2 do not have any common factors, which means the lowest common denominator of the rational expressions is the quantity x minus 3 times the quantity x minus 2. Rewrite each rational expression with the least common denominator. Multiply the numerator and denominator of 3 divided by the quantity x minus 3 by the quantity x minus 2. Multiply the numerator and denominator of 2 divided by the quantity x minus 2 by the quantity x minus 2. The result is the rational expression 3 times the quantity x minus 2 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression 2 times the quantity x minus 3 divided by the quantity x minus 2 times the quantity x minus 3. Simplify the numerators and keep the denominators factored. The numerator of the first rational expression, 3 times the quantity x minus 2, simplifies to 3 x minus 6. The numerator of the second rational expression, 2 times the quantity x minus 3, simplifies to 2 x minus 6. The result is the rational expression the quantity 3 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression, the quantity 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2.

Step 2 is to add or subtract the rational expressions by adding the numerators, the quantity 3 x minus 6 and the quantity 2 x minus 6, and placing the sum over the denominator, the quantity x minus 3 times the quantity x minus 2. The result is the quantity 3 x minus 6 plus 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2. Simplify the numerator by combining like terms. The result is the quantity 5 x minus 12 all divided by the quantity x minus 3 times the quantity x minus 2.

Step 3. Notice that 5 x minus 12 cannot be factored, so the answer is simplified.

Try It #9

Add: $\frac{2}{x-2}+\frac{5}{x+3}.$

$\frac{7x-4}{(x-2)(x+3)}$

Did you get it?

Try It #10

Add: $\frac{4}{m+3}+\frac{3}{m+4}.$

$\frac{7m+25}{(m+3)(m+4)}$

Did you get it?

The steps used to add rational expressions are summarized here.

Add or subtract rational expressions.

Determine if the expressions have a common denominator.
- Yes – go to step 2.
- No – Rewrite each rational expression with the LCD.
  - Find the LCD.
  - Rewrite each rational expression as an equivalent rational expression with the LCD.
Add or subtract the rational expressions.
Simplify, if possible.

Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as $\frac{3x-6}{(x-3)(x-2)}$ to be able to add it to $\frac{2x-6}{(x-2)(x-3)}.$ Simplify only after you have combined the numerators.

Example 6

Add: $\frac{8}{{x}^{2}-2x-3}+\frac{3x}{{x}^{2}+4x+3}.$

Factor each denominator to find the LCD, then rewrite both fractions over it before adding.

Table 7
	$A mathematical expression showing the sum of two algebraic fractions: 8 over (x^2 - 2x - 3) plus 3x over (x^2 + 4x + 3).$
Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
$\begin{array}{llll} \\ \\ \text{Find the LCD.} & & & \begin{array}{l}\,{x}^{2}-2x-3=(x+1)(x-3) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{{x}^{2}+4x+3=(x+1)\,(x+3)}\, \\ \\ \,\text{LCD}\,=(x+1)(x-3)(x+3)\end{array}\end{array}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.	$A fraction with the numerator 3x^2 - x + 24 and the denominator (x + 1)(x - 3)(x + 3).$
Simplify the numerator.	$A fraction with the numerator 3x^2 - x + 24 and the denominator (x + 1)(x - 3)(x + 3).$
	The numerator is prime, so there are no common factors.

Try It #11

Add: $\frac{1}{{m}^{2}-m-2}+\frac{5m}{{m}^{2}+3m+2}.$

$\frac{5{m}^{2}-9m+2}{(m+1)(m-2)(m+2)}$

Did you get it?

Try It #12

Add: $\frac{2n}{{n}^{2}-3n-10}+\frac{6}{{n}^{2}+5n+6}.$

$\frac{2{n}^{2}+12n-30}{(n+2)(n-5)(n+3)}$

Did you get it?

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

Example 7

Subtract: $\frac{8y}{{y}^{2}-16}-\frac{4}{y-4}.$

Factor the first denominator to find the LCD, then rewrite the second fraction over it.

Table 8
	$A mathematical expression shows a subtraction of two fractions: 8y over the quantity y squared minus 16, minus 4 over the quantity y minus 4.$
Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
$\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}{y}^{2}-16=(y-4)(y+4) \\ \,\underset{\_\_\_\_\_\_\_\_\_\_\_\_}{\,y-4\,=y-4} \\ \,\text{LCD}\,=(y-4)(y+4)\end{array}\end{array}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.	$An algebraic fraction featuring (8y - 4y - 16) in the numerator and (y - 4)(y + 4) in the denominator.$
Simplify the numerator.	$A mathematical expression displaying a fraction. The numerator is '4y - 16', and the denominator is '(y - 4)(y + 4)'.$
Factor the numerator to look for common factors.	$A mathematical fraction shown with the expression 4(y-4) in the numerator and (y-4)(y+4) in the denominator. This represents an algebraic expression that can be simplified.$
Remove common factors	$A mathematical expression showing the cancellation of the term (y-4) in both the numerator 4(y-4) and the denominator (y-4)(y+4) of a fraction during simplification.$
Simplify.	$A fraction with a numerator of 4 and a denominator of (y + 4).$

Try It #13

Subtract: $\frac{2x}{{x}^{2}-4}-\frac{1}{x+2}.$

$\frac{1}{x-2}$

Did you get it?

Try It #14

Subtract: $\frac{3}{z+3}-\frac{6z}{{z}^{2}-9}.$

$\frac{-3}{z-3}$

Did you get it?

There are lots of negative signs in the next example. Be extra careful.

Example 8

Subtract: $\frac{-3n-9}{{n}^{2}+n-6}-\frac{n+3}{2-n}.$

Factor the first denominator, then notice n-2 and 2-n are opposites and multiply the second fraction by -1/-1.

Table 9
	$A mathematical expression showing the subtraction of two algebraic fractions. The first fraction is (-3n-9)/(n^2+n-6) and the second is (n+3)/(2-n).$
Factor the denominator.	$A mathematical expression featuring two fractions separated by a minus sign. The first fraction is (-3n - 9) divided by (n - 2)(n + 3). The second fraction is (n + 3) divided by (2 - n).$
Since $n-2$ and $2-n$ are opposites, we will multiply the second rational expression by $\frac{-1}{-1}.$
	$A mathematical expression showing the subtraction of two algebraic fractions. The first fraction is (-3n - 9) divided by ((n - 2)(n + 3)), and the second is (-1)(n + 3) divided by (n - 2).$
Simplify. Remember, $a-(\text{-}b)=a+b.$	$A mathematical expression showing the sum of two algebraic fractions: (-3n - 9) / ((n - 2)(n + 3)) + (n + 3) / (n - 2).$
Do the rational expressions have a common denominator? No.
$\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}{n}^{2}+n-6=(n-2)(n+3) \\ \,\underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{n-2=(n-2)} \\ \,\text{LCD}\,=(n-2)(n+3)\end{array}\end{array}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.	$An algebraic fraction with a numerator of -3n - 9 + n^2 + 6n + 9 and a denominator of (n-2)(n+3).$
Simplify the numerator.	$A mathematical expression showing the fraction (n^2 + 3n) / ((n - 2)(n + 3)).$
Factor the numerator to look for common factors.
Simplify.	$A mathematical expression displaying the fraction n divided by the quantity (n-2).$

Try It #15

Subtract : $\frac{3x-1}{{x}^{2}-5x-6}-\frac{2}{6-x}.$

$\frac{5x+1}{(x-6)(x+1)}$

Did you get it?

Try It #16

Subtract: $\frac{-2y-2}{{y}^{2}+2y-8}-\frac{y-1}{2-y}.$

$\frac{y+3}{y+4}$

Did you get it?

Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.

Example 9

Subtract: $\frac{4}{{a}^{2}+6a+5}-\frac{3}{{a}^{2}+7a+10}.$

Factor both denominators to find the LCD, then rewrite each fraction over it before subtracting.

Table 10
	$A mathematical expression featuring two fractions subtracted from each other: 4 divided by (a squared plus 6a plus 5) minus 3 divided by (a squared plus 7a plus 10).$
Factor the denominators.	$A mathematical expression showing the subtraction of two algebraic fractions. The first fraction is 4 over (a+1)(a+5), and the second is 3 over (a+2)(a+5).$
Do the rational expressions have a common denominator? No.
$\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}\,{a}^{2}+6a+5\,=(a+1)(a+5) \\ \,\underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{{a}^{2}+7a+10\,=\,(a+5)(a+2)} \\ \,\text{LCD}=(a+1)(a+5)(a+2)\end{array}\end{array}$
Rewrite each rational expression as an equivalent rational expression with the LCD.	$A mathematical expression showing the subtraction of two algebraic fractions. The first fraction is 4(a+2) over (a+1)(a+5)(a+2), and the second is 3(a+1) over (a+2)(a+5)(a+1).$
Simplify the numerators.	$A mathematical expression displaying the subtraction of two algebraic fractions: (4a + 8)/((a + 1)(a + 5)(a + 2)) - (3a + 3)/((a + 2)(a + 5)(a + 1)).$
Subtract the rational expressions.	$A mathematical fraction with the numerator '4a + 8 - (3a - 3)' and the denominator '(a + 1)(a + 5)(a + 2)'.$
Simplify the numerator.	$A mathematical fraction with the numerator '4a + 8 - 3a + 3' and the denominator '(a + 1)(a + 5)(a + 2)'.$
	$A fraction with numerator (a+5) and denominator (a+1)(a+5)(a+2).$
Look for common factors.	$An algebraic fraction with (a+5) in the numerator and (a+1)(a+5)(a+2) in the denominator, showing (a+5) being canceled out from both the numerator and the denominator.$
Simplify.	$A mathematical fraction displaying 1 in the numerator and the product of (a+1) and (a+2) in the denominator.$

Try It #17

Subtract: $\frac{3}{{b}^{2}-4b-5}-\frac{2}{{b}^{2}-6b+5}.$

$\frac{1}{(b+1)(b-1)}$

Did you get it?

Try It #18

Subtract: $\frac{4}{{x}^{2}-4}-\frac{3}{{x}^{2}-x-2}.$

$\frac{1}{(x+2)(x+1)}$

Did you get it?

We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to find their LCD.

Example 10

Simplify: $\frac{2u}{u-1}+\frac{1}{u}-\frac{2u-1}{{u}^{2}-u}.$

Factor all three denominators first so you can find one LCD that works for every term.

Table 11

Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
$\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}u-1\,=\,(u-1) \\ u\,=u \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{{u}^{2}-u=u(u-1)} \\ \text{LCD}\,=u(u-1)\end{array}\end{array}$
Rewrite each rational expression as an equivalent rational expression with the LCD.	$A multi-term algebraic expression featuring fractions with a common denominator, u(u-1). The expression is (2u * u) / ((u - 1)u) + (1 * (u - 1)) / (u * (u - 1)) - (2u - 1) / (u(u - 1)), with some 'u' and '(u-1)' terms highlighted in red.$

Write as one rational expression.	$A mathematical fraction with numerator 2u^2 + u - 1 - 2u + 1 and denominator u(u - 1).$
Simplify.	$A mathematical expression presented as a fraction: (2u^2 - u) divided by u(u - 1).$
Factor the numerator, and remove common factors.	$A mathematical expression showing a fraction with 'u-dot(2u-1)' in the numerator and 'u-dot(u-1)' in the denominator, set against a plain white background.$
Simplify.	$A mathematical expression displaying a fraction with (2u - 1) in the numerator and (u - 1) in the denominator.$

Try It #19

Simplify: $\frac{v}{v+1}+\frac{3}{v-1}-\frac{6}{{v}^{2}-1}.$

$\frac{v+3}{v+1}$

Did you get it?

Try It #20

Simplify: $\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{{w}^{2}+9w+14}.$

$\frac{3w}{w+7}$

Did you get it?

Add and subtract rational functions

To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.

Example 11

Find $R(x)=f(x)-g(x)$ where $f(x)=\frac{x+5}{x-2}$ and $g(x)=\frac{5x+18}{{x}^{2}-4}.$

Substitute f(x) and g(x), then factor the denominators to find their LCD.

Table 12

Substitute in the functions $f(x),$ $g(x).$
Factor the denominators.
Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
$\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}x-2\,=(x-2) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{{x}^{2}-4=(x-2)(x+2)} \\ \,\text{LCD}=(x-2)(x+2)\end{array}\end{array}$
Rewrite each rational expression as an equivalent rational expression with the LCD.	$An algebraic expression showing R(x) as the difference of two rational functions with a common denominator (x-2)(x+2). The numerator of the first fraction is (x+5)(x+2) and the second is 5x+18.$
Write as one rational expression.
Simplify.

Factor the numerator, and remove common factors.	$A mathematical equation shows R(x) as a fraction with the numerator (x + 4) times (x - 2) and the denominator (x - 2) times (x + 2), where the (x - 2) terms are crossed out in both the numerator and the denominator.$
Simplify.	$A mathematical expression for R(x) is shown, where R(x) is equal to the fraction (x + 4) divided by (x + 2).$

Try It #21

Find $R(x)=f(x)-g(x)$ where $f(x)=\frac{x+1}{x+3}$ and $g(x)=\frac{x+17}{{x}^{2}-x-12}.$

$\frac{x-7}{x-4}$

Did you get it?

Try It #22

Find $R(x)=f(x)+g(x)$ where $f(x)=\frac{x-4}{x+3}$ and $g(x)=\frac{4x+6}{{x}^{2}-9}.$

$\frac{{x}^{2}-3x+18}{(x+3)(x-3)}$

Did you get it?

Media

Access this online resource for additional instruction and practice with adding and subtracting rational expressions.

Add and Subtract Rational Expressions- Unlike Denominators

Key Concepts

Rational Expression Addition and Subtraction
If p, q, and r are polynomials where $r\ne 0,$ then
$\,\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}$ and $\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}$ See Example 1, Example 2, and Example 3.
How to find the least common denominator of rational expressions.
- Factor each expression completely.
- List the factors of each expression. Match factors vertically when possible.
- Bring down the columns.
- Write the LCD as the product of the factors.
See Example 4.
How to add or subtract rational expressions.
- Determine if the expressions have a common denominator.
  - Yes – go to step 2.
  - No – Rewrite each rational expression with the LCD.
    - Find the LCD.
    - Rewrite each rational expression as an equivalent rational expression with the LCD.
- Add or subtract the rational expressions.
- Simplify, if possible.
See Example 5, Example 6, Example 7, Example 8, Example 9, Example 10, and Example 11.

Section Exercises

Practice Makes Perfect

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

$\frac{2}{15}+\frac{7}{15}$

$\frac{3}{5}$

$\frac{7}{24}+\frac{11}{24}$

$\frac{3c}{4c-5}+\frac{5}{4c-5}$

$\frac{3c+5}{4c-5}$

$\frac{7m}{2m+n}+\frac{4}{2m+n}$

$\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}$

$r+8$

$\frac{3{s}^{2}}{3s-2}+\frac{13s-10}{3s-2}$

$\frac{2{w}^{2}}{{w}^{2}-16}+\frac{8w}{{w}^{2}-16}$

$\frac{2w}{w-4}$

$\frac{7{x}^{2}}{{x}^{2}-9}+\frac{21x}{{x}^{2}-9}$

In the following exercises, subtract.

$\frac{9{a}^{2}}{3a-7}-\frac{49}{3a-7}$

$3a+7$

$\frac{25{b}^{2}}{5b-6}-\frac{36}{5b-6}$

$\frac{3{m}^{2}}{6m-30}-\frac{21m-30}{6m-30}$

$\frac{m-2}{2}$

$\frac{2{n}^{2}}{4n-32}-\frac{18n-16}{4n-32}$

$\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}$

$\frac{p+3}{p+5}$

$\frac{5{q}^{2}+3q-9}{{q}^{2}+6q+8}-\frac{4{q}^{2}+9q+7}{{q}^{2}+6q+8}$

$\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\frac{4{r}^{2}+5r+30}{{r}^{2}-49}$

$\frac{r+9}{r+7}$

$\frac{7{t}^{2}-t-4}{{t}^{2}-25}-\frac{6{t}^{2}+12t-44}{{t}^{2}-25}$

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

$\frac{10v}{2v-1}+\frac{2v+4}{1-2v}$

$4$

$\frac{20w}{5w-2}+\frac{5w+6}{2-5w}$

$\frac{10{x}^{2}+16x-7}{8x-3}+\frac{2{x}^{2}+3x-1}{3-8x}$

$x+2$

$\frac{6{y}^{2}+2y-11}{3y-7}+\frac{3{y}^{2}-3y+17}{7-3y}$

$\frac{{z}^{2}+6z}{{z}^{2}-25}-\frac{3z+20}{25-{z}^{2}}$

$\frac{z+4}{z-5}$

$\frac{{a}^{2}+3a}{{a}^{2}-9}-\frac{3a-27}{9-{a}^{2}}$

$\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\frac{2{b}^{2}-5b-8}{49-{b}^{2}}$

$\frac{4b-3}{b-7}$

$\frac{{c}^{2}+5c-10}{{c}^{2}-16}-\frac{{c}^{2}-8c-10}{16-{c}^{2}}$

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

$\frac{5}{{x}^{2}-2x-8},\frac{2x}{{x}^{2}-x-12}$

ⓐ $(x+2)(x-4)(x+3)$
ⓑ $\frac{5x+15}{(x+2)(x-4)(x+3)}$ ,
$\frac{2{x}^{2}+4x}{(x+2)(x-4)(x+3)}$

$\frac{8}{{y}^{2}+12y+35},\frac{3y}{{y}^{2}+y-42}$

$\frac{9}{{z}^{2}+2z-8},\frac{4z}{{z}^{2}-4}$

ⓐ $(z-2)(z+4)(z+2)$
ⓑ $\frac{9z+18}{(z-2)(z+4)(z+2)}$ ,
$\frac{4{z}^{2}+16z}{(z-2)(z+4)(z+2)}$

$\frac{6}{{a}^{2}+14a+45},\frac{5a}{{a}^{2}-81}$

$\frac{4}{{b}^{2}+6b+9},\frac{2b}{{b}^{2}-2b-15}$

ⓐ $(b+3)(b+3)(b-5)$
ⓑ $\frac{4b-20}{(b+3)(b+3)(b-5)}$ ,
$\frac{2{b}^{2}+6b}{(b+3)(b+3)(b-5)}$

$\frac{5}{{c}^{2}-4c+4},\frac{3c}{{c}^{2}-7c+10}$

$\frac{2}{3{d}^{2}+14d-5},\frac{5d}{3{d}^{2}-19d+6}$

ⓐ $(d+5)(3d-1)(d-6)$
ⓑ $\frac{2d-12}{(d+5)(3d-1)(d-6)}$ ,
$\frac{5{d}^{2}+25d}{(d+5)(3d-1)(d-6)}$

$\frac{3}{5{m}^{2}-3m-2},\frac{6m}{5{m}^{2}+17m+6}$

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

$\frac{7}{10{x}^{2}y}+\frac{4}{15x{y}^{2}}$

$\frac{21y+8x}{30{x}^{2}{y}^{2}}$

$\frac{1}{12{a}^{3}{b}^{2}}+\frac{5}{9{a}^{2}{b}^{3}}$

$\frac{3}{r+4}+\frac{2}{r-5}$

$\frac{5r-7}{(r+4)(r-5)}$

$\frac{4}{s-7}+\frac{5}{s+3}$

$\frac{5}{3w-2}+\frac{2}{w+1}$

$\frac{11w+1}{(3w-2)(w+1)}$

$\frac{4}{2x+5}+\frac{2}{x-1}$

$\frac{2y}{y+3}+\frac{3}{y-1}$

$\frac{2{y}^{2}+y+9}{(y+3)(y-1)}$

$\frac{3z}{z-2}+\frac{1}{z+5}$

$\frac{5b}{{a}^{2}b-2{a}^{2}}+\frac{2b}{{b}^{2}-4}$

$\frac{b(5b+10+2{a}^{2})}{{a}^{2}(b-2)(b+2)}$

$\frac{4}{cd+3c}+\frac{1}{{d}^{2}-9}$

$\frac{-3m}{3m-3}+\frac{5m}{{m}^{2}+3m-4}$

$\text{-}\frac{m}{m+4}$

$\frac{8}{4n+4}+\frac{6}{{n}^{2}-n-2}$

$\frac{3r}{{r}^{2}+7r+6}+\frac{9}{{r}^{2}+4r+3}$

$\frac{3({r}^{2}+6r+18)}{(r+1)(r+6)(r+3)}$

$\frac{2s}{{s}^{2}+2s-8}+\frac{4}{{s}^{2}+3s-10}$

$\frac{t}{t-6}-\frac{t-2}{t+6}$

$\frac{2(7t-6)}{(t-6)(t+6)}$

$\frac{x-3}{x+6}-\frac{x}{x+3}$

$\frac{5a}{a+3}-\frac{a+2}{a+6}$

$\frac{4{a}^{2}+25a-6}{(a+3)(a+6)}$

$\frac{3b}{b-2}-\frac{b-6}{b-8}$

$\frac{6}{m+6}-\frac{12m}{{m}^{2}-36}$

$\frac{-6}{m-6}$

$\frac{4}{n+4}-\frac{8n}{{n}^{2}-16}$

$\frac{-9p-17}{{p}^{2}-4p-21}-\frac{p+1}{7-p}$

$\frac{p+2}{p+3}$

$\frac{-13q-8}{{q}^{2}+2q-24}-\frac{q+2}{4-q}$

$\frac{-2r-16}{{r}^{2}+6r-16}-\frac{5}{2-r}$

$\frac{3}{r-2}$

$\frac{2t-30}{{t}^{2}+6t-27}-\frac{2}{3-t}$

$\frac{2x+7}{10x-1}+3$

$\frac{4(8x+1)}{10x-1}$

$\frac{8y-4}{5y+2}-6$

$\frac{3}{{x}^{2}-3x-4}-\frac{2}{{x}^{2}-5x+4}$

$\frac{x-5}{(x-4)(x+1)(x-1)}$

$\frac{4}{{x}^{2}-6x+5}-\frac{3}{{x}^{2}-7x+10}$

$\frac{5}{{x}^{2}+8x-9}-\frac{4}{{x}^{2}+10x+9}$

$\frac{1}{(x-1)(x+1)}$

$\frac{3}{2{x}^{2}+5x+2}-\frac{1}{2{x}^{2}+3x+1}$

$\frac{5a}{a-2}+\frac{9}{a}-\frac{2a+18}{{a}^{2}-2a}$

$\frac{5{a}^{2}+7a-36}{a(a-2)}$

$\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2{b}^{2}-10b}$

$\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{{c}^{2}-4}$

$\frac{c-5}{c+2}$

$\frac{6d}{d-5}+\frac{1}{d+4}-\frac{7d-5}{{d}^{2}-d-20}$

$\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{{d}^{2}+2d}$

$\frac{3(d+1)}{d+2}$

$\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{{q}^{2}+2q-15}$

Add and Subtract Rational Functions

In the following exercises, find ⓐ $R(x)=f(x)+g(x)$ ⓑ $R(x)=f(x)-g(x).$

$f(x)=\frac{-5x-5}{{x}^{2}+x-6}$ and
$\,g(x)=\frac{x+1}{2-x}$

ⓐ $R(x)=-\frac{(x+8)(x+1)}{(x-2)(x+3)}$ ⓑ $R(x)=\frac{x+1}{x+3}$

$f(x)=\frac{-4x-24}{{x}^{2}+x-30}$ and
$\,g(x)=\frac{x+7}{5-x}$

$f(x)=\frac{6x}{{x}^{2}-64}$ and
$\,g(x)=\frac{3}{x-8}$

ⓐ $\frac{3(3x+8)}{(x-8)(x+8)}$
ⓑ $R(x)=\frac{3}{x+8}$

$f(x)=\frac{5}{x+7}$ and
$\,g(x)=\frac{10x}{{x}^{2}-49}$

Writing Exercises

Donald thinks that $\frac{3}{x}+\frac{4}{x}$ is $\frac{7}{2x}.$ Is Donald correct? Explain.

Answers will vary.

Explain how you find the Least Common Denominator of ${x}^{2}+5x+4$ and ${x}^{2}-16.$

Felipe thinks $\frac{1}{x}+\frac{1}{y}$ is $\frac{2}{x+y}.$ ⓐ Choose numerical values for x and y and evaluate $\frac{1}{x}+\frac{1}{y}.$ ⓑ Evaluate $\frac{2}{x+y}$ for the same values of x and y you used in part ⓐ. ⓒ Explain why Felipe is wrong. ⓓ Find the correct expression for $\frac{1}{x}+\frac{1}{y}.$

Simplify the expression $\frac{4}{{n}^{2}+6n+9}-\frac{1}{{n}^{2}-9}$ and explain all your steps.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and six rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

	\(\,\frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4},\,x\ne \text{-}4\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\,\frac{11x+28+{x}^{2}}{x+4}\)
Write the degrees in descending order.	\(\,\frac{{x}^{2}+11x+28}{x+4}\)
Factor the numerator.	\(\,\frac{(x+4)(x+7)}{x+4}\)
Simplify by removing common factors.	\(\,\frac{(x+4)(x+7)}{x+4}\)
Simplify.	\(\,x+7\)

	\(\,\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
Subtract the numerators and place the difference over the common denominator.	\(\,\frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}\)
Distribute the sign in the numerator.	\(\,\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\)
Combine like terms.	\(\,\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\)
Factor the numerator and the denominator.	\(\,\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
Simplify by removing common factors.	\(\,\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
	\(\,\frac{(x-2)}{(x+3)}\)

	$A mathematical expression showing the sum of two fractions: 7 over d plus 5 over negative d.$
Multiply the second fraction by \(\frac{-1}{-1}.\)	$A mathematical expression showing the sum of two fractions. The first fraction is 7 divided by d. The second fraction has a numerator of (-1) multiplied by 5, and a denominator of (-1) multiplied by (-d).$
The denominators are the same.	$A mathematical expression showing the addition of two fractions with a common denominator 'd'. The expression is 7/d + (-5)/d.$
Simplify.	$A mathematical fraction displaying the number 2 over the letter d, indicating 2 divided by d, centered on a white background.$

Find the LCD for \(\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.\)
Factor each denominator completely, lining up common factors. Bring down the columns.
Write the LCD as the product of the factors.


Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}u-1\,=\,(u-1) \\ u\,=u \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{{u}^{2}-u=u(u-1)} \\ \text{LCD}\,=u(u-1)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.	$A multi-term algebraic expression featuring fractions with a common denominator, u(u-1). The expression is (2u * u) / ((u - 1)u) + (1 * (u - 1)) / (u * (u - 1)) - (2u - 1) / (u(u - 1)), with some 'u' and '(u-1)' terms highlighted in red.$

Write as one rational expression.	$A mathematical fraction with numerator 2u^2 + u - 1 - 2u + 1 and denominator u(u - 1).$
Simplify.	$A mathematical expression presented as a fraction: (2u^2 - u) divided by u(u - 1).$
Factor the numerator, and remove common factors.	$A mathematical expression showing a fraction with 'u-dot(2u-1)' in the numerator and 'u-dot(u-1)' in the denominator, set against a plain white background.$
Simplify.	$A mathematical expression displaying a fraction with (2u - 1) in the numerator and (u - 1) in the denominator.$


Substitute in the functions \(f(x),\) \(g(x).\)
Factor the denominators.
Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
\(\begin{array}{llll}\text{Find the LCD.} & & & \begin{array}{l}x-2\,=(x-2) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{{x}^{2}-4=(x-2)(x+2)} \\ \,\text{LCD}=(x-2)(x+2)\end{array}\end{array}\)
Rewrite each rational expression as an equivalent rational expression with the LCD.	$An algebraic expression showing R(x) as the difference of two rational functions with a common denominator (x-2)(x+2). The numerator of the first fraction is (x+5)(x+2) and the second is 5x+18.$
Write as one rational expression.
Simplify.

Factor the numerator, and remove common factors.	$A mathematical equation shows R(x) as a fraction with the numerator (x + 4) times (x - 2) and the denominator (x - 2) times (x + 2), where the (x - 2) terms are crossed out in both the numerator and the denominator.$
Simplify.	$A mathematical expression for R(x) is shown, where R(x) is equal to the fraction (x + 4) divided by (x + 2).$