Add Rational Expressions With Like Denominators

Adding rational expressions with like denominators

When two rational expressions have the same denominator, you add them the same way you add fractions with a common denominator: add the numerators and keep the denominator.

Method

1. Check the denominators are identical.
2. Add the numerators carefully, distributing signs if needed.
3. Write the result over the common denominator.
4. Simplify the numerator by combining like terms.
5. Reduce the fraction if possible by factoring and canceling any common factor.

Example pattern

If [ \frac{a}{d}+\frac{b}{d}, ] then [ \frac{a+b}{d}. ] If the numerators are expressions, such as polynomials, combine like terms before finishing.

Important checks

• Make sure you do not add the denominators.
• Watch for subtraction: (\frac{a}{d}+\frac{-b}{d}=\frac{a-b}{d}).
• If the final numerator and denominator share a factor, simplify the expression.

Quick check

You can verify your answer by substituting a value that does not make the denominator zero. Compute each original expression and your simplified result; they should match.

This skill is mainly about keeping the common denominator unchanged while combining the top parts correctly and then simplifying the final expression.