8.3: Add and Subtract Rational Expressions with a Common Denominator
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 Lynn Marecek
 Professor (Mathematics) at Santa Ana College
 Publisher: OpenStax CNX
 Add rational expressions with a common denominator
 Subtract rational expressions with a common denominator
 Add and subtract rational expressions whose denominators are opposites
Before you get started, take this readiness quiz.
If you miss a problem, go back to the section listed and review the material.
 Add: \(\frac{y}{3}+\frac{9}{3}\).
If you missed this problem, review [link].  Subtract: \(\frac{10}{x}−\frac{2}{x}\).
If you missed this problem, review [link].  Factor completely: \(8n^5−20n^3\).
If you missed this problem, review [link].  Factor completely: \(45a^3−5ab^2\).
If you missed this problem, review [link].
Add 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.
Definition: RATIONAL EXPRESSION ADDITION
If p, q, and r are polynomials where \(r \ne 0\), then
\(\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\)
To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
We will add two numerical fractions first, to remind us of how this is done.
Example \(\PageIndex{1}\)
Add: \(\frac{5}{18}+\frac{7}{18}\).
 Answer

\(\frac{5}{18}+\frac{7}{18}\) The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{5+7}{18}\) Add in the numerator. \(\frac{12}{18}\) Factor the numerator and denominator to show the common factors. \(\frac{6·2}{6·3}\) Simplify. \(\frac{2}{3}\)
Example \(\PageIndex{2}\)
Add: \(\frac{7}{16}+\frac{5}{16}\).
 Answer

\(\frac{3}{4}\)
Example \(\PageIndex{3}\)
Add: \(\frac{3}{10}+\frac{1}{10}\).
 Answer

\(\frac{2}{5}\)
Example \(\PageIndex{4}\)
Add: \(\frac{3y}{4y−3}+\frac{7}{4y−3}\).
 Answer

\(\frac{3y}{4y−3}+\frac{7}{4y−3}\). The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{3y+7}{4y−3}\) The numerator and denominator cannot be factored. The fraction is simplified.
Example \(\PageIndex{5}\)
Add: \(\frac{5x}{2x+3}+\frac{2}{2x+3}\).
 Answer

\(\frac{5x+2}{2x+3}\).
Example \(\PageIndex{6}\)
Add: \(\frac{x}{x−2}+\frac{1}{x−2}\).
 Answer

\(\frac{x+1}{x−2}\)
Example \(\PageIndex{7}\)
Add: \(\frac{7x+12}{x+3}+\frac{x^2}{x+3}\).
 Answer

\(\frac{7x+12}{x+3}+\frac{x^2}{x+3}\) The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{7x+12+x^2}{x+3}\) Write the degrees in descending order. \(\frac{x^2+7x+12}{x+3}\) Factor the numerator. \(\frac{(x+3)(x+4)}{x+3}\) Simplify. x+4
Example \(\PageIndex{8}\)
Add: \(\frac{9x+14}{x+7}+\frac{x^2}{x+7}\).
 Answer

x+2
Example \(\PageIndex{9}\)
Add: \(\frac{x^2+8x}{x+5}+\frac{15}{x+5}\).
 Answer

x+3
Subtract Rational Expressions with a Common Denominator
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.
Definition: RATIONAL EXPRESSION SUBTRACTION
If p, q, and r are polynomials where \(r \ne 0\)
\(\frac{p}{r}−\frac{q}{r}=\frac{p−q}{r}\)
To subtract rational expressions, subtract the numerators and place the difference over the common denominator.
Example \(\PageIndex{10}\)
Subtract: \(\frac{n^2}{n−10}−\frac{100}{n−10}\).
 Answer

\(\frac{n^2}{n−10}−\frac{100}{n−10}\) The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{n^2−100}{n−10}\) Factor the numerator. \(\frac{(n−10)(n+10)}{n−10}\) Simplify. n+10
Example \(\PageIndex{11}\)
Subtract: \(\frac{x^2}{x+3}−\frac{9}{x+3}\).
 Answer

x−3
Example \(\PageIndex{12}\)
Subtract: \(\frac{4x^2}{2x−5}−\frac{25}{2x−5}\).
 Answer

2x+5
Be careful of the signs when you subtract a binomial!
Example \(\PageIndex{13}\)
Subtract: \(\frac{y^2}{y−6}−\frac{2y+24}{y−6}\).
 Answer

\(\frac{y^2}{y−6}−\frac{2y+24}{y−6}\) The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{y^2−(2y+24)}{y−6}\) Distribute the sign in the numerator. \(\frac{y^2−2y−24}{y−6}\) Factor the numerator. \(\frac{(y−6)(y+4)}{y−6}\) Simplify. y+4
Example \(\PageIndex{14}\)
Subtract: \(\frac{n^2}{n−4}−\frac{n+12}{n−4}\).
 Answer

n+3
Example \(\PageIndex{15}\)
Subtract: \(\frac{y^2}{y−1}−\frac{9y−8}{y−1}\).
 Answer

y−8
Example \(\PageIndex{16}\)
Subtract: \(\frac{5x^2−7x+3}{x^2−3x+18}−\frac{4x^2+x−9}{x^2−3x+18}\).
 Answer

\(\frac{5x^2−7x+3}{x^2−3x+18}−\frac{4x^2+x−9}{x^2−3x+18}\) The fractions have a common denominator, so add the numerators and place the sum over the common denominator. \(\frac{5x^2−7x+3−(4x^2+x−9)}{x^2−3x+18}\) Distribute the sign in the numerator. \(\frac{5x^2−7x+3−4x^2−x+9}{x^2−3x+18}\) 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. \(\frac{x−2}{x+3}\)
Example \(\PageIndex{17}\)
Subtract: \(\frac{4x^2−11x+8}{x^2−3x+2}−\frac{3x^2+x−3}{x^2−3x+2}\).
 Answer

\(\frac{x−11}{x−2}\)
Example \(\PageIndex{18}\)
Subtract: \(\frac{6x^2−x+20}{x^2−81}−\frac{5x^2+11x−7}{x^2−81}\).
 Answer

\(\frac{x−3}{x+9}\)
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.
Multiply the second fraction by \(\frac{−1}{−1}\).  
The denominators are the same.  
Simplify. 
Example \(\PageIndex{19}\)
Add: \(\frac{4u−1}{3u−1}+\frac{u}{1−3u}\).
 Answer

Multiply the second fraction by \(\frac{−1}{−1}\). Simplify the second fraction. The denominators are the same. Add the numerators. Simplify. Simplify.
Example \(\PageIndex{20}\)
Add: \(\frac{8x−15}{2x−5}+\frac{2x}{5−2x}\).
 Answer

3
Example \(\PageIndex{21}\)
Add: \(\frac{6y^2+7y−10}{4y−7}+\frac{2y^2+2y+11}{7−4y}\).
 Answer

y+3
Example \(\PageIndex{22}\)
Subtract: \(\frac{m^2−6m}{m^2−1}−\frac{3m+2}{1−m^2}\).
 Answer

Multiply the second fraction by \(\frac{−1}{−1}\). Simplify the second fraction. The denominators are the same. Subtract the numerators. Distribute. m2−6m+3m+2m2−1 Combine like terms. Factor the numerator and denominator. Simplify by removing common factors. Simplify.
Example \(\PageIndex{23}\)
Subtract: \(\frac{y^2−5y}{y^2−4}−\frac{6y−6}{4−y^2}\).
 Answer

\(\frac{y+3}{y+2}\)
Example \(\PageIndex{24}\)
Subtract: \(\frac{2n^2+8n−1}{n^2−1}−\frac{n^2−7n−1}{1−n^2}\).
 Answer

\(\frac{3n−2}{n−1}\)
Key Concepts
 Rational Expression Addition

If p, q, and r are polynomials where \(r \ne 0\), then
\(\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\)
 To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

 Rational Expression Subtraction

If p, q, and r are polynomials where \(r \ne 0\)
\(\frac{p}{r}−\frac{q}{r}=\frac{p−q}{r}\)
 To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Practice Makes Perfect
Add Rational Expressions with a Common Denominator
In the following exercises, add.
Example \(\PageIndex{25}\)
\(\frac{2}{15}+\frac{7}{15}\)
 Answer

\(\frac{3}{5}\)
Example \(\PageIndex{26}\)
\(\frac{4}{21}+\frac{3}{21}\)
Example \(\PageIndex{27}\)
\(\frac{7}{24}+\frac{11}{24}\)
 Answer

\(\frac{3}{4}\)
Example \(\PageIndex{28}\)
\(\frac{7}{36}+\frac{13}{36}\)
Example \(\PageIndex{29}\)
\(\frac{3a}{a−b}+\frac{1}{a−b}\)
 Answer

\(\frac{3a+1}{a+b}\)
Example \(\PageIndex{30}\)
\(\frac{3c}{4c−5}+\frac{5}{4c−5}\)
Example \(\PageIndex{31}\)
\(\frac{d}{d+8}+\frac{5}{d+8}\)
 Answer

\(\frac{d+5}{d+8}\)
Example \(\PageIndex{32}\)
\(\frac{7m}{2m+n}+\frac{4}{2m+n}\)
Example \(\PageIndex{33}\)
\(\frac{p^2+10p}{p+2}+\frac{16}{p+2}\)
 Answer

p+8
Example \(\PageIndex{34}\)
\(\frac{q^2+12q}{q+3}+\frac{27}{q+3}\)
Example \(\PageIndex{35}\)
\(\frac{2r^2}{2r−1}+\frac{15r−8}{2r−1}\)
 Answer

r+8
Example \(\PageIndex{36}\)
\(\frac{3s^2}{3s−2}+\frac{13s−10}{3s−2}\)
Example \(\PageIndex{37}\)
\(\frac{8t^2}{t+4}+\frac{32t}{t+4}\)
 Answer

8t
Example \(\PageIndex{38}\)
\(\frac{6v^2}{v+5}+\frac{30v}{v+5}\)
Example \(\PageIndex{39}\)
\(\frac{2w^2}{w^2−16}+\frac{8w}{w^2−16}\)
 Answer

\(\frac{2w}{w−4}\)
Example \(\PageIndex{40}\)
\(\frac{7x^2}{x^2−9}+\frac{21x}{x^2−9}\)
In the following exercises, subtract.
Example \(\PageIndex{41}\)
\(\frac{y^2}{y+8}−\frac{64}{y+8}\)
 Answer

y−8
Example \(\PageIndex{42}\)
\(\frac{z^2}{z+2}−\frac{4}{z+2}\)
Example \(\PageIndex{43}\)
\(\frac{9a^2}{3a−7}−\frac{49}{3a−7}\)
 Answer

3a+7
Example \(\PageIndex{44}\)
\(\frac{25b^2}{5b−6}−\frac{36}{5b−6}\)
Example \(\PageIndex{45}\)
\(\frac{c^2}{c−8}−\frac{6c+16}{c−8}\)
 Answer

c+2
Example \(\PageIndex{46}\)
\(\frac{d^2}{d−9}−\frac{6d+27}{d−9}\)
Example \(\PageIndex{47}\)
\(\frac{3m^2}{6m−30}−\frac{21m−30}{6m−30}\)
 Answer

\(\frac{m−2}{3}\)
Example \(\PageIndex{48}\)
\(\frac{2n^2}{4n−32}−\frac{30n−16}{4n−32}\)
Example \(\PageIndex{49}\)
\(\frac{6p^2+3p+4}{p^2+4p−5}−\frac{5p^2+p+7}{p^2+4p−5}\)
 Answer

\(\frac{p+3}{p+5}\)
Example \(\PageIndex{50}\)
\(\frac{5q^2+3q−9}{q^2+6q+8}−\frac{4q^2+9q+7}{q^2+6q+8}\)
Example \(\PageIndex{51}\)
\(\frac{5r^2+7r−33}{r^2−49}−\frac{4r^2−5r−30}{r^2−49}\)
 Answer

\(\frac{r+9}{r+7}\)
Example \(\PageIndex{52}\)
\(\frac{7t^2−t−4}{t^2−25}−\frac{6t^2+2t−1}{t^2−25}\)
In the following exercises, add.
Example \(\PageIndex{53}\)
\(\frac{10v^2}{v−1}+\frac{2v+4}{1−2v}\)
 Answer

4
Example \(\PageIndex{54}\)
\(\frac{20w}{5w−2}+\frac{5w+6}{2−5w}\)
Example \(\PageIndex{55}\)
\(\frac{10x^2+16x−7}{8x−3}+\frac{2x^2+3x−1}{3−8x}\)
 Answer

x+2
Example \(\PageIndex{56}\)
\(\frac{6y^2+2y−11}{3y−7}+\frac{3y^2−3y+17}{7−3y}\)
In the following exercises, subtract.
Example \(\PageIndex{57}\)
\(\frac{z^2+6z}{z^2−25}−\frac{3z+20}{25−z^2}\)
 Answer

\(\frac{z+4}{z−5}\)
Example \(\PageIndex{58}\)
\(\frac{a^2+3a}{a^2−9}−\frac{3a−27}{9−a^2}\)
Example \(\PageIndex{59}\)
\(\frac{2b^2+30b−13}{b^2−49}−\frac{2b^2−5b−8}{49−b^2}\)
 Answer

\(\frac{4b−3}{b−7}\)
Example \(\PageIndex{60}\)
\(\frac{c^2+5c−10}{c^2−16}−\frac{c^2−8c−10}{16−c^2}\)
Everyday Math
Example \(\PageIndex{61}\)
Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If rr represents Sarah’s speed when she ran, then her running time is modeled by the expression \(\frac{8}{r}\) and her biking time is modeled by the expression \(\frac{24}{r+4}\). Add the rational expressions \(\frac{8}{r}+\frac{24}{r+4}\) to get an expression for the total amount of time Sarah ran and biked.
 Answer

\(\frac{32r+32}{r(r+4)}\)
Example \(\PageIndex{62}\)
If Pete can paint a wall in pp hours, then in one hour he can paint \(\frac{1}{p}\) of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint \(\frac{1}{p+3}\) of the wall. Add the rational expressions \(\frac{1}{p}+\frac{1}{p+3}\) to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.
Writing Exercises
Example \(\PageIndex{63}\)
Donald thinks that \(\frac{3}{x}+\frac{4}{x}\) is \(\frac{7}{2x}\). Is Donald correct? Explain.
Example \(\PageIndex{64}\)
Explain how you find the Least Common Denominator of \(x^2+5x+4\) and \(x^2−16\).
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?