Adding and Subtracting Rational Expressions Formula

The Formula

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} (or use the LCD for simpler results)

When to use: Just like \frac{1}{3} + \frac{1}{4} requires a common denominator of 12, adding \frac{2}{x+1} + \frac{3}{x-2} requires the LCD (x+1)(x-2). Rewrite each fraction so both have the same denominator, then add the numerators. The process mirrors numeric fractions but with polynomial denominators.

Quick Example

\frac{2}{x+1} + \frac{3}{x-2} = \frac{2(x-2) + 3(x+1)}{(x+1)(x-2)} = \frac{5x - 1}{(x+1)(x-2)}

Notation

LCD is the Least Common Denominator. Each fraction is rewritten with the LCD before combining numerators.

What This Formula Means

Adding or subtracting rational expressions by finding a least common denominator (LCD), rewriting each fraction with the LCD, then combining the numerators over the common denominator.

Just like \frac{1}{3} + \frac{1}{4} requires a common denominator of 12, adding \frac{2}{x+1} + \frac{3}{x-2} requires the LCD (x+1)(x-2). Rewrite each fraction so both have the same denominator, then add the numerators. The process mirrors numeric fractions but with polynomial denominators.

Formal View

\frac{P}{Q} + \frac{R}{S} = \frac{PS + RQ}{QS}. Using \mathrm{LCD} = \mathrm{lcm}(Q, S): rewrite as \frac{P \cdot (\mathrm{LCD}/Q) + R \cdot (\mathrm{LCD}/S)}{\mathrm{LCD}}. The LCD is the product of all irreducible factors at their highest multiplicities.

Worked Examples

Example 1

medium
Add \frac{2}{x+1} + \frac{3}{x-2}.

Solution

  1. 1
    Step 1: LCD = (x+1)(x-2).
  2. 2
    Step 2: \frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x+1)(x-2)}.
  3. 3
    Step 3: Combine: \frac{2x - 4 + 3x + 3}{(x+1)(x-2)} = \frac{5x - 1}{(x+1)(x-2)}.
  4. 4
    Check: At x = 3: \frac{2}{4} + \frac{3}{1} = \frac{7}{2} and \frac{14}{4} = \frac{7}{2} ✓

Answer

\frac{5x - 1}{(x+1)(x-2)}
To add rational expressions with different denominators, find the LCD, rewrite each fraction with the LCD, then combine numerators. This mirrors adding numeric fractions.

Example 2

hard
Subtract \frac{x}{x+2} - \frac{3}{x^2 + 4x + 4}.

Common Mistakes

  • Using the product of denominators instead of the LCD—this works but creates unnecessarily large expressions
  • Forgetting to distribute when multiplying the numerator by the missing LCD factors
  • Not simplifying the final answer by factoring and canceling common factors from the combined numerator and the LCD

Why This Formula Matters

Adding and subtracting rational expressions is essential for solving rational equations, partial fraction decomposition in calculus, and combining algebraic fractions in applied problems.

Frequently Asked Questions

What is the Adding and Subtracting Rational Expressions formula?

Adding or subtracting rational expressions by finding a least common denominator (LCD), rewriting each fraction with the LCD, then combining the numerators over the common denominator.

How do you use the Adding and Subtracting Rational Expressions formula?

Just like \frac{1}{3} + \frac{1}{4} requires a common denominator of 12, adding \frac{2}{x+1} + \frac{3}{x-2} requires the LCD (x+1)(x-2). Rewrite each fraction so both have the same denominator, then add the numerators. The process mirrors numeric fractions but with polynomial denominators.

What do the symbols mean in the Adding and Subtracting Rational Expressions formula?

LCD is the Least Common Denominator. Each fraction is rewritten with the LCD before combining numerators.

Why is the Adding and Subtracting Rational Expressions formula important in Math?

Adding and subtracting rational expressions is essential for solving rational equations, partial fraction decomposition in calculus, and combining algebraic fractions in applied problems.

What do students get wrong about Adding and Subtracting Rational Expressions?

Finding the LCD when denominators are polynomials that need factoring. Factor each denominator first, then build the LCD from all unique factors at their highest powers.

What should I learn before the Adding and Subtracting Rational Expressions formula?

Before studying the Adding and Subtracting Rational Expressions formula, you should understand: simplifying rational expressions, least common multiple.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Rational Expressions: Simplifying, Operations, and Domain Restrictions →
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