Lesson 8.3: Add and Subtract Rational Expressions with a Common Denominator

New Concept

Learn to add and subtract rational expressions, just like with numerical fractions. When denominators match, simply combine the numerators. For opposite denominators, a quick multiplication by $\frac{-1}{-1}$ creates a common base for solving.

What’s next

Soon, you'll master these rules through interactive examples and a series of practice cards designed to build your skills step-by-step.

Property

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

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator. Always check if the resulting fraction can be simplified.

Examples

• Add: $\frac{5x}{x+2} + \frac{10}{x+2}$.

This equals $\frac{5x+10}{x+2}$. Factoring the numerator gives $\frac{5(x+2)}{x+2}$, which simplifies to $5$.

• Add: $\frac{x^2}{x-5} + \frac{2x-35}{x-5}$.

This equals $\frac{x^2+2x-35}{x-5}$. Factoring the numerator gives $\frac{(x+7)(x-5)}{x-5}$, which simplifies to $x+7$.

Property

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

To subtract rational expressions, subtract the numerators and place the difference over the common denominator. Be very careful to distribute the negative sign to every term in the second numerator.

Examples

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

This equals $\frac{y^2-16}{y-4}$. Factoring the numerator as a difference of squares gives $\frac{(y-4)(y+4)}{y-4}$, which simplifies to $y+4$.

• Subtract: $\frac{x^2}{x+5} - \frac{3x+10}{x+5}$.

This equals $\frac{x^2 - (3x+10)}{x+5} = \frac{x^2-3x-10}{x+5}$. Factoring the numerator gives $\frac{(x-5)(x+2)}{x+5}$. This cannot be simplified further.

Property

When the denominators of two rational expressions are opposites (e.g., $x-2$ and $2-x$), a common denominator can be created by multiplying one of the fractions by $\frac{-1}{-1}$. This changes the sign of its denominator to match the other, while also changing the sign of its numerator. For example, $\frac{a}{x-2} + \frac{b}{2-x}$ becomes $\frac{a}{x-2} - \frac{b}{x-2}$.

Examples

• Add: $\frac{6x-5}{5x-2} + \frac{x-3}{2-5x}$.

Multiply the second fraction by $\frac{-1}{-1}$ to get $\frac{6x-5}{5x-2} - \frac{x-3}{5x-2} = \frac{6x-5-(x-3)}{5x-2} = \frac{5x-2}{5x-2} = 1$.

• Subtract: $\frac{k^2-10k}{k^2-9} - \frac{2k+21}{9-k^2}$.

Multiply the second fraction by $\frac{-1}{-1}$ to get $\frac{k^2-10k}{k^2-9} + \frac{2k+21}{k^2-9} = \frac{k^2-8k+21}{k^2-9}$. This cannot be simplified.