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Adding and Subtracting Rational Expressions

Jul 1, 2025

Overview

This lecture covers how to add and subtract similar fractions and rational algebraic expressions, including those with unlike denominators, using methods such as finding the least common denominator (LCD) and the butterfly method.

Adding and Subtracting Similar Fractions

  • To add or subtract similar fractions, find the LCD of the denominators.
  • Divide the LCD by each denominator and multiply by the corresponding numerator.
  • Add or subtract the resulting numerators, then simplify if possible.
  • The butterfly method involves cross-multiplying and adding/subtracting results to get a common denominator.

Adding and Subtracting Rational Expressions with Unlike Denominators

  • Begin by completely factoring all denominators.
  • Write each denominator in exponential notation to find their factors.
  • The LCD is determined by taking the highest degree of each factor from the denominators.
  • Express each rational expression with the new LCD as denominator, adjusting numerators accordingly.
  • Add or subtract the numerators and keep the LCD as the denominator.

Examples

  • Example 1: (\frac{2}{2} + \frac{1}{4}) uses LCD 4 or the butterfly method to yield (\frac{5}{4}) or (1 \frac{1}{4}).
  • Example 2: (\frac{2}{2} - \frac{1}{6}) uses LCD 6, yielding (\frac{5}{6}), or the butterfly method then simplifying to (\frac{5}{6}).
  • Example 3: Add (\frac{5}{8m^2 n^4} + \frac{2}{6m^3 n}) with LCD (24m^3 n^4), resulting in (\frac{15m + 8n^3}{24m^3 n^4}).
  • Example 4: Add (\frac{2y}{5x^2} + \frac{3x}{4xy}) with LCD (20x^2y), resulting in (\frac{8y^2 + 15x^2}{20x^2y}).
  • Example 5: Subtract (\frac{4x}{x^2 - 25} - \frac{5}{x-5}) with LCD ((x + 5)(x - 5)), result is (\frac{-x - 25}{(x + 5)(x - 5)}).

Key Terms & Definitions

  • LCD (Least Common Denominator) — the smallest common multiple of all denominators in the expressions.
  • Butterfly Method — a shortcut involving cross-multiplication to get a common denominator for addition/subtraction.
  • Rational Algebraic Expression — a fraction where the numerator and/or denominator are polynomials.

Action Items / Next Steps

  • Practice similar problems by finding LCDs and rewriting expressions.
  • Review factoring polynomials and recognizing the difference of squares.
  • Complete assigned homework on adding and subtracting rational algebraic expressions.

Full transcript