Lesson 7.2: Add and Subtract Rational Expressions

New Concept

This lesson extends fraction arithmetic to rational expressions. You'll master adding and subtracting them by finding a common denominator, just like with numbers, tackling common, opposite, and unlike denominators to simplify expressions and functions.

What’s next

Next, you’ll tackle worked examples for adding and subtracting with common denominators, then apply your skills in a series of interactive practice cards.

Property

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

To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator. Be sure to factor and simplify the final result if possible.

Examples

• Add: $\frac{7x+10}{x+2} + \frac{x^2}{x+2}$. This becomes $\frac{x^2+7x+10}{x+2}$. Factoring the numerator gives $\frac{(x+2)(x+5)}{x+2}$, which simplifies to $x+5$.

• Subtract: $\frac{6x^2-2x+1}{x^2-4x-5} - \frac{5x^2+3x-4}{x^2-4x-5}$. This becomes $\frac{6x^2-2x+1-(5x^2+3x-4)}{x^2-4x-5} = \frac{x^2-5x+5}{x^2-4x-5}$.

Property

When the denominators of two rational expressions are opposites (e.g., $d$ and $-d$, or $x-2$ 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, without changing the fraction's value.

Examples

• Subtract: $\frac{y^2-8y}{y^2-9} - \frac{2y+3}{9-y^2}$. We multiply the second fraction by $\frac{-1}{-1}$ to get $\frac{y^2-8y}{y^2-9} - \frac{-1(2y+3)}{-1(9-y^2)} = \frac{y^2-8y-(-2y-3)}{y^2-9} = \frac{y^2-6y+3}{y^2-9}$.

• Add: $\frac{9}{k-3} + \frac{2}{3-k}$. This becomes $\frac{9}{k-3} + \frac{-2}{k-3} = \frac{7}{k-3}$.

Property

To find the least common denominator (LCD) of rational expressions:
Step 1. Factor each denominator completely.
Step 2. List the factors of each denominator. Match factors vertically when possible.
Step 3. Bring down the columns by including all factors, but do not include common factors more than the maximum number of times they appear in any single factorization.
Step 4. Write the LCD as the product of these factors.

Examples

• For $\frac{6}{x^2-9}$ and $\frac{2x}{x^2-x-6}$, the denominators factor to $(x-3)(x+3)$ and $(x-3)(x+2)$. The LCD is $(x-3)(x+3)(x+2)$.

• For $\frac{3}{a^2+6a+9}$ and $\frac{5a}{a^2-9}$, the denominators factor to $(a+3)(a+3)$ and $(a-3)(a+3)$. The LCD is $(a+3)^2(a-3)$.

Property

To add or subtract rational expressions with unlike denominators:
Step 1. Determine if the expressions have a common denominator. If not, find the LCD. Then, rewrite each rational expression as an equivalent expression with the LCD.
Step 2. Add or subtract the numerators and place the result over the common denominator.
Step 3. Simplify the resulting rational expression, if possible.

Examples

• Add: $\frac{5}{x-4} + \frac{3}{x+2}$. The LCD is $(x-4)(x+2)$. This becomes $\frac{5(x+2)}{(x-4)(x+2)} + \frac{3(x-4)}{(x-4)(x+2)} = \frac{5x+10+3x-12}{(x-4)(x+2)} = \frac{8x-2}{(x-4)(x+2)}$.

• Subtract: $\frac{7y}{y^2-25} - \frac{2}{y-5}$. The LCD is $(y-5)(y+5)$. This becomes $\frac{7y}{(y-5)(y+5)} - \frac{2(y+5)}{(y-5)(y+5)} = \frac{7y-2y-10}{(y-5)(y+5)} = \frac{5y-10}{(y-5)(y+5)}$.

Property

To add or subtract rational functions, such as finding $R(x) = f(x) + g(x)$ or $R(x) = f(x) - g(x)$, you perform the operation on their corresponding expressions. The techniques are the same as adding or subtracting any rational expressions: find the LCD, rewrite the expressions, combine, and simplify.

Examples

• Find $R(x) = f(x) - g(x)$ where $f(x) = \frac{x+4}{x-3}$ and $g(x) = \frac{x+18}{x^2-9}$. The LCD is $(x-3)(x+3)$. $R(x) = \frac{(x+4)(x+3)}{(x-3)(x+3)} - \frac{x+18}{(x-3)(x+3)} = \frac{x^2+7x+12-x-18}{(x-3)(x+3)} = \frac{x^2+6x-6}{(x-3)(x+3)}$.

• Find $R(x) = f(x) + g(x)$ where $f(x) = \frac{x}{x+1}$ and $g(x) = \frac{2}{x-2}$. The LCD is $(x+1)(x-2)$. $R(x) = \frac{x(x-2)}{(x+1)(x-2)} + \frac{2(x+1)}{(x+1)(x-2)} = \frac{x^2-2x+2x+2}{(x+1)(x-2)} = \frac{x^2+2}{x^2-x-2}$.