8.3: Add and Subtract Rational Expressions with a Common Denominator
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 Contributed by 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.
