8.3 Add and Subtract Rational Expressions with a Common Denominator

Adding and Subtracting Rational Expressions with a Common Denominator

Rational expressions are fractions with polynomials in the numerator and denominator. When two rational expressions share the same denominator, you can add or subtract them by combining the numerators, just like you would with numeric fractions. The key skill here is combining those numerators correctly and then simplifying the result.

Introduction to Rational Expressions

A rational expression is the quotient of two polynomials, like $\frac{x+3}{x^2-1}$. The denominator can never equal zero, since division by zero is undefined.

For this section, you're working with rational expressions that already have a common denominator, meaning the denominator polynomial is identical in each expression. When the denominators match, the process is straightforward. (Finding a common denominator when they don't match is a separate skill covered later.)

Addition of Rational Expressions

When two rational expressions have the same denominator, add them by adding the numerators and keeping the denominator the same.

$\frac{A}{C} + \frac{B}{C} = \frac{A + B}{C}$

Example 1:

$\frac{2}{x} + \frac{3}{x} = \frac{2+3}{x} = \frac{5}{x}$

Example 2 (with polynomial numerators):

$\frac{2x+3}{y} + \frac{x-1}{y} = \frac{(2x+3)+(x-1)}{y} = \frac{3x+2}{y}$

Notice in Example 2 that you combine like terms in the numerator after adding. Always check whether the numerator can be simplified further.

Subtraction of Rational Expressions

Subtraction works the same way, but you subtract the entire second numerator. This is where sign errors happen most often. Distribute the negative sign across every term in the second numerator.

$\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}$

Example 3:

$\frac{6x}{9} - \frac{2x}{9} = \frac{6x - 2x}{9} = \frac{4x}{9}$

Example 4 (where the negative must be distributed):

$\frac{x^2 - 4}{x+2} - \frac{3x + 6}{x+2} = \frac{(x^2 - 4) - (3x + 6)}{x+2} = \frac{x^2 - 3x - 10}{x+2}$

That negative sign applies to both the $3x$ and the $+6$, giving you $-3x - 6$. Forgetting to distribute is the most common mistake in this section.

Simplifying the Result

After combining the numerators, always check whether you can simplify by factoring.

1. Combine the numerators over the common denominator.
2. Simplify the numerator (combine like terms, expand if needed).
3. Factor both the numerator and denominator.
4. Cancel any common factors.

Continuing Example 4:

$\frac{x^2 - 3x - 10}{x+2} = \frac{(x-5)(x+2)}{x+2} = x - 5$

The $(x+2)$ factor cancels from the numerator and denominator. Your final answer is $x - 5$ (with the restriction that $x \neq -2$, since the original expression is undefined there).

Don't skip the factoring step. Even if the numerator looks messy, try factoring it. That's often where the real simplification happens.

Opposite Denominators

Sometimes two expressions look like they have different denominators, but the denominators are actually opposites (like $x$ and $-x$, or $a - b$ and $b - a$). You can handle these by recognizing that $\frac{A}{-C} = \frac{-A}{C}$. Rewriting one expression this way gives you a common denominator.

Example 5 (sum with opposite denominators):

$\frac{3}{x} + \frac{3}{-x} = \frac{3}{x} + \frac{-3}{x} = \frac{3 - 3}{x} = 0$

Example 6 (difference with opposite denominators):

$\frac{2}{y} - \frac{2}{-y} = \frac{2}{y} - \frac{-2}{y} = \frac{2 + 2}{y} = \frac{4}{y}$

The general pattern: factor out $-1$ from one denominator, move the negative sign to the numerator, and then you have a common denominator to work with.