Add and Subtract Rational Expressions

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.

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?

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.

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} .

| {: valign=”top”}| The denominators are the same. |

| {: valign=”top”}| Simplify. |

| {: valign=”top”}{: .unnumbered .unstyled summary=”7 divided by d plus 5 divided by negative d. Multiply 5 divided by d times the fraction, negative 1 over negative 1. The result is 7 divided by d plus the quantity negative 1 times 5 all divided by the quantity negative 1times negative d. Notice that the denominators are the same. The denominators are d. 7 divided by d plus negative 5 divided by d. Simplify by adding the numerators. The result is 2 divided by d.” data-label=””}

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

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.

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.

Remember, we always exclude values that would make the denominator zero. What values of

x

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.

How to Add Rational Expressions with Unlike Denominators

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

![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.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_07_02_006a_img.jpg) ![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.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_07_02_006b_img.jpg) ![Step 3. Notice that 5 x minus 12 cannot be factored, so the answer is simplified.](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_07_02_006c_img.jpg)

Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as

\frac{3 x - 6}{(x - 3) (x - 2)}

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.

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

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.

Add and subtract rational functions

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

Key Concepts

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

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

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

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

r + 8

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

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

\frac{2 w}{w - 4}

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

In the following exercises, subtract.

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

3 a + 7

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

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

\frac{m - 2}{2}

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

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

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

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

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

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

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

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

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

4

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

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

x + 2

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

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

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

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

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

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

\frac{c^{2} + 5 c - 10}{c^{2} - 16} - \frac{c^{2} - 8 c - 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} - 2 x - 8}, \frac{2 x}{x^{2} - x - 12}

ⓐ $(x + 2) (x - 4) (x + 3)$

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

,* * *

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

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

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

ⓐ $(z - 2) (z + 4) (z - 4)$

ⓑ $\frac{9 z - 36}{(z - 2) (z + 4) (z - 4)}$

,* * *

\frac{4 z^{2} - 8 z}{(z - 2) (z + 4) (z - 4)}

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

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

ⓐ $(b + 3) (b + 3) (b - 5)$

ⓑ $\frac{4 b - 20}{(b + 3) (b + 3) (b - 5)}$

,* * *

\frac{2 b^{2} + 6 b}{(b + 3) (b + 3) (b - 5)}

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

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

ⓐ $(d + 5) (3 d - 1) (d - 6)$

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

,* * *

\frac{5 d^{2} + 25 d}{(d + 5) (3 d - 1) (d - 6)}

\frac{3}{5 m^{2} - 3 m - 2}, \frac{6 m}{5 m^{2} + 17 m + 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}{15 x y^{2}}

\frac{21 y + 8 x}{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{5 r - 7}{(r + 4) (r - 5)}

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

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

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

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

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

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

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

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

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

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

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

− \frac{m}{m + 4}

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

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

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

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

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

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

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

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

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

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

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

\frac{−6}{m - 6}

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

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

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

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

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

\frac{3}{r - 2}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\frac{2 q}{q + 5} + \frac{3}{q - 3} - \frac{13 q + 15}{q^{2} + 2 q - 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{−5 x - 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{−4 x - 24}{x^{2} + x - 30}

and* * *

g (x) = \frac{x + 7}{5 - x}

f (x) = \frac{6 x}{x^{2} - 64}

and* * *

g (x) = \frac{3}{x - 8}

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

ⓑ $R (x) = \frac{3}{x + 8}$

f (x) = \frac{5}{x + 7}

and* * *

g (x) = \frac{10 x}{x^{2} - 49}

Writing Exercises

Donald thinks that $\frac{3}{x} + \frac{4}{x}$

is $\frac{7}{2 x} .$

Is Donald correct? Explain.

Answers will vary.

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

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 ⓐ.* * *

ⓓ Find the correct expression for $\frac{1}{x} + \frac{1}{y} .$

ⓐ Answers will vary.* * *

ⓑ Answers will vary.* * *

ⓓ $\frac{x + y}{x y}$

Simplify the expression $\frac{4}{n^{2} + 6 n + 9} - \frac{1}{n^{2} - 9}$

and explain all your steps.

Self Check

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

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


The denominators are opposites, so multiply the second fraction by $\frac{−1}{−1} .$
Simplify the second fraction.
The denominators are the same. Subtract the numerators.
Distribute.
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.


Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
$\begin{matrix} Find the LCD. & \begin{matrix} x^{2} - 2 x - 3 = (x + 1) (x - 3) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{x^{2} + 4 x + 3 = (x + 1) (x + 3)} \\ LCD = (x + 1) (x - 3) (x + 3) \end{matrix} \end{matrix}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
	The numerator is prime, so there are no common factors.


Do the expressions have a common denominator?	No.
Rewrite each expression with the LCD.
$\begin{matrix} Find the LCD. & \begin{matrix} y^{2} - 16 = (y - 4) (y + 4) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_}{ y - 4 = y - 4} \\ LCD = (y - 4) (y + 4) \end{matrix} \end{matrix}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Remove common factors
Simplify.


Factor the denominator.
Since $n - 2$ and $2 - n$ are opposites, we will multiply the second rational expression by $\frac{−1}{−1} .$

Simplify. Remember, $a - (− b) = a + b .$
Do the rational expressions have a common denominator? No.
$\begin{matrix} Find the LCD. & \begin{matrix} n^{2} + n - 6 = (n - 2) (n + 3) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{n - 2 = (n - 2)} \\ LCD = (n - 2) (n + 3) \end{matrix} \end{matrix}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Add the rational expressions.
Simplify the numerator.
Factor the numerator to look for common factors.
Simplify.


Factor the denominators.
Do the rational expressions have a common denominator? No.
$\begin{matrix} Find the LCD. & \begin{matrix} a^{2} + 6 a + 5 = (a + 1) (a + 5) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{a^{2} + 7 a + 10 = (a + 5) (a + 2)} \\ LCD = (a + 1) (a + 5) (a + 2) \end{matrix} \end{matrix}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerator.

Look for common factors.
Simplify.


{: valign=”top”}	Multiply the second fraction by $\frac{−1}{−1} .$

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


Factor each denominator.
Multiply each denominator by the ‘missing’ LCD factor and multiply each numerator by the same factor.
Simplify the numerators.


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.
$\begin{matrix} Find the LCD. & \begin{matrix} x - 2 = (x - 2) \\ \underset{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}{x^{2} - 4 = (x - 2) (x + 2)} \\ LCD = (x - 2) (x + 2) \end{matrix} \end{matrix}$
Rewrite each rational expression as an equivalent rational expression with the LCD.
Write as one rational expression.
Simplify.

Factor the numerator, and remove common factors.
Simplify.