Add and Subtract Rational Expressions with a Common Denominator

By the end of this section, you will be able to: add rational expressions with a common denominator, subtract rational expressions with a common denominator, and add and subtract rational expressions whose denominators are opposites.

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, 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. If $p$ , $q$ , and $r$ are polynomials where $r \neq 0$ , 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. Add $\tfrac{5}{18} + \tfrac{7}{18}$ .

The fractions have a common denominator, so add the numerators and place the sum over the common denominator. Then factor and remove common factors to simplify:

\frac{5}{18} + \frac{7}{18} = \frac{5 + 7}{18} = \frac{12}{18} = \frac{6 \cdot 2}{6 \cdot 3} = \frac{2}{3}

Now we’ll add rational expressions the same way.

Example. Add $\tfrac{3y}{4y - 3} + \tfrac{7}{4y - 3}$ .

The fractions have a common denominator, so add the numerators and place the sum over the common denominator:

\frac{3y}{4y - 3} + \frac{7}{4y - 3} = \frac{3y + 7}{4y - 3}

The numerator and denominator cannot be factored, so the fraction is simplified.

Remember, we do not allow values that would make the denominator zero. What value of $y$ should be excluded here? The value $y = \tfrac{3}{4}$ , because it makes $4y - 3$ equal to zero.

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

When we add rational expressions, the sum often factors, letting us simplify.

Example. Add $\tfrac{7x + 12}{x + 3} + \tfrac{x^2}{x + 3}$ .

The fractions have a common denominator, so add the numerators and place the sum over the common denominator. Write the numerator in descending order, factor it, then remove the common factor:

\begin{array}{rcl} \tfrac{7x + 12}{x + 3} + \tfrac{x^2}{x + 3} &=& \tfrac{7x + 12 + x^2}{x + 3} \\[6pt] &=& \tfrac{x^2 + 7x + 12}{x + 3} \\[6pt] &=& \tfrac{(x + 3)(x + 4)}{x + 3} \\[6pt] &=& x + 4 \end{array}

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.

Rational Expression Subtraction. If $p$ , $q$ , and $r$ are polynomials where $r \neq 0$ , then

\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.

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

Example. Subtract $\tfrac{n^2}{n - 10} - \tfrac{100}{n - 10}$ .

The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator. Factor the numerator, then remove the common factor:

\begin{array}{rcl} \tfrac{n^2}{n - 10} - \tfrac{100}{n - 10} &=& \tfrac{n^2 - 100}{n - 10} \\[6pt] &=& \tfrac{(n - 10)(n + 10)}{n - 10} \\[6pt] &=& n + 10 \end{array}

Be careful of the signs when you subtract a binomial!

Example. Subtract $\tfrac{y^2}{y - 6} - \tfrac{2y + 24}{y - 6}$ .

Subtract the numerators over the common denominator. Because we are subtracting a binomial, distribute the sign so both terms change. Then combine like terms, factor, and remove the common factor:

\begin{array}{rcl} \tfrac{y^2}{y - 6} - \tfrac{2y + 24}{y - 6} &=& \tfrac{y^2 - (2y + 24)}{y - 6} \\[6pt] &=& \tfrac{y^2 - 2y - 24}{y - 6} \\[6pt] &=& \tfrac{(y - 6)(y + 4)}{y - 6} \\[6pt] &=& y + 4 \end{array}

Notice how distributing the sign changed $-(2y + 24)$ into $-2y - 24$ . Missing that step is the most common mistake when subtracting rational expressions.

Subtract: $\tfrac{n^2}{n - 4} - \tfrac{n + 12}{n - 4}$ .

The numerators can be larger polynomials, but the process is the same: subtract, distribute the sign, combine like terms, then factor and simplify.

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

Subtract the numerators over the common denominator, distribute the sign, combine like terms, then factor and remove the common factor:

\begin{array}{rcl} \tfrac{5x^2 - 7x + 3}{x^2 - 3x - 18} - \tfrac{4x^2 + x - 9}{x^2 - 3x - 18} &=& \tfrac{5x^2 - 7x + 3 - (4x^2 + x - 9)}{x^2 - 3x - 18} \\[6pt] &=& \tfrac{5x^2 - 7x + 3 - 4x^2 - x + 9}{x^2 - 3x - 18} \\[6pt] &=& \tfrac{x^2 - 8x + 12}{x^2 - 3x - 18} \\[6pt] &=& \tfrac{(x - 2)(x - 6)}{(x + 3)(x - 6)} \\[6pt] &=& \tfrac{x - 2}{x + 3} \end{array}

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 $\tfrac{-1}{-1}$ .

Let’s see how this works with the numerical-looking example $\tfrac{7}{d} + \tfrac{5}{-d}$ . Multiply the second fraction by $\tfrac{-1}{-1}$ so both denominators become $d$ , then add:

\frac{7}{d} + \frac{5}{-d} = \frac{7}{d} + \frac{(-1)5}{(-1)(-d)} = \frac{7}{d} + \frac{-5}{d} = \frac{2}{d}

Example. Add $\tfrac{4u - 1}{3u - 1} + \tfrac{u}{1 - 3u}$ .

The denominators $3u - 1$ and $1 - 3u$ are opposites, so multiply the second fraction by $\tfrac{-1}{-1}$ . That turns its denominator into $3u - 1$ ; now the denominators match, so add the numerators and simplify:

\begin{array}{rcl} \tfrac{4u - 1}{3u - 1} + \tfrac{u}{1 - 3u} &=& \tfrac{4u - 1}{3u - 1} + \tfrac{(-1)u}{(-1)(1 - 3u)} \\[6pt] &=& \tfrac{4u - 1}{3u - 1} + \tfrac{-u}{3u - 1} \\[6pt] &=& \tfrac{4u - 1 - u}{3u - 1} \\[6pt] &=& \tfrac{3u - 1}{3u - 1} \\[6pt] &=& 1 \end{array}

The same idea works for subtraction. Multiply the fraction with the opposite denominator by $\tfrac{-1}{-1}$ first, and then subtract as usual — watching the signs.

Example. Subtract $\tfrac{m^2 - 6m}{m^2 - 1} - \tfrac{3m + 2}{1 - m^2}$ .

The denominators $m^2 - 1$ and $1 - m^2$ are opposites, so multiply the second fraction by $\tfrac{-1}{-1}$ . Then the denominators match; subtract the numerators, distribute the sign, combine like terms, factor, and simplify:

\begin{array}{rcl} \tfrac{m^2 - 6m}{m^2 - 1} - \tfrac{3m + 2}{1 - m^2} &=& \tfrac{m^2 - 6m}{m^2 - 1} - \tfrac{-(3m + 2)}{m^2 - 1} \\[6pt] &=& \tfrac{m^2 - 6m}{m^2 - 1} - \tfrac{-3m - 2}{m^2 - 1} \\[6pt] &=& \tfrac{m^2 - 6m - (-3m - 2)}{m^2 - 1} \\[6pt] &=& \tfrac{m^2 - 6m + 3m + 2}{m^2 - 1} \\[6pt] &=& \tfrac{m^2 - 3m + 2}{m^2 - 1} \\[6pt] &=& \tfrac{(m - 1)(m - 2)}{(m - 1)(m + 1)} \\[6pt] &=& \tfrac{m - 2}{m + 1} \end{array}

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

Key terms

common denominator — a single denominator shared by two or more rational expressions, allowing their numerators to be added or subtracted directly. opposite denominators — denominators that are negatives of each other (such as $3u - 1$ and $1 - 3u$ ); multiplying one fraction by $\tfrac{-1}{-1}$ turns them into a common denominator.

This section is adapted from Elementary Algebra 2e, Section 8.3: Add and Subtract Rational Expressions with a Common Denominator by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked examples as prose with display equality chains, condensed the numerical warm-up into the addition rule, and folded the excluded-value discussion into the addition example; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Multiply and Divide Rational Expressions Add and Subtract Rational Expressions with Unlike Denominators