Lesson 4: Adding and Subtracting Rational Expressions

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.