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Adding and Subtracting Rational Algebraic Expressions

Jun 14, 2024

Lecture Notes on Adding and Subtracting Rational Algebraic Expressions

Adding and Subtracting Similar Fractions

Steps:

  1. Find the Least Common Denominator (LCD).
  2. Solve for the numerator.
  3. Simplify the expression.

Example 1: Adding Fractions

  • Given: (\frac{2}{2} + \frac{1}{4})
  • Steps:
    • LCD: 4
    • Calculation: ( 4 \div 2 = 2 \text{ (times 2)} = 4 4 \div 4 = 1 \text{ (times 1)} = 1 4 + 1 = 5 )
    • Result: (\frac{5}{4}) or (1 \frac{1}{4})

Butterfly Method

  • Multiply the denominators and cross-multiply the numerators:
    • LCD: (8)
    • Calculation:
      • Numerator: (2 \times 4 = 8, 1 \times 2 = 2)
      • Sum: (8 + 2 = 10)
    • Result: (\frac{10}{8}) or (1 \frac{1}{4})

Example 2: Subtracting Fractions

  • Given: (\frac{2}{2} - \frac{1}{6})
  • Steps:
    • LCD: 6
    • Calculation: ( 6 \div 2 = 3 \text{ (times 2)} = 6 6 \div 6 = 1 \text{ (times 1)} = 1 3 - 1 = 2 )
    • Result: (\frac{5}{6})

Butterfly Method

  • Multiply the denominators and cross-multiply the numerators:
    • LCD: (12)
    • Calculation:
      • Numerator: (2 \times 6 = 12, 1 \times 2 = 2)
      • Difference: (12 - 2 = 10)
    • Result: (\frac{10}{12}) or (\frac{5}{6})

Adding and Subtracting Rational Algebraic Expressions with Unlike Denominators

Steps:

  1. Factor the denominators completely.
  2. Find the LCD.
  3. Write equivalent expressions using the LCD.
  4. Add or subtract the numerators.
  5. Simplify and keep the LCD.

Example 1: Adding Rational Expressions

  • Given: (\frac{5}{8m^2n^4} + \frac{2}{6m^3n})
  • Steps:
    • Factor Denominators:
      • (8m^2n^4 = 2^3m^2n^4)
      • (6m^3n = 2 \times 3 \times m^3 \times n)
    • Find LCD: Highest degree of each factor.
      • Factors: (2^3, 3, m^3, n^4)
      • LCD: (24m^3n^4)
    • Write Equivalent Expressions:
      • (\frac{5 \times 3m}{24m^3n^4} + \frac{2 \times 4n^3}{24m^3n^4})
      • Simplify Numerators: (15m + 8n^3)
    • Result: (\frac{15m + 8n^3}{24m^3n^4})

Example 2: Adding Rational Expressions

  • Given: (\frac{2y}{5x^2} + \frac{3x}{4xy})
  • Steps:
    • Factor Denominators:
      • (5x^2)
      • (4xy)
    • Find LCD: Highest degree of each factor.
      • Factors: (5, 4, x^2, y)
      • LCD: (20x^2y)
    • Write Equivalent Expressions:
      • (\frac{2y \times 4}{20x^2y} + \frac{3x \times 5}{20x^2y})
      • Simplify Numerators: (8y^2 + 15x^2)
    • Result: (\frac{8y^2 + 15x^2}{20x^2y})

Example 3: Subtracting Rational Expressions

  • Given: (\frac{4x}{x^2 - 25} - \frac{5}{x - 5})
  • Steps:
    • Factor Denominators:
      • (\frac{4x}{(x+5)(x-5)} - \frac{5}{x-5})
    • Find LCD: ((x+5)(x-5))
    • Write Equivalent Expressions:
      • (\frac{4x}{(x+5)(x-5)} - \frac{5(x+5)}{(x+5)(x-5)})
      • Simplify Numerators: (4x - [5(x + 5)] = 4x - 5x - 25 = -x - 25)
    • Result: (\frac{-x - 25}{(x+5)(x-5)})

Conclusion:

  • Practice finding the LCD, writing equivalent fractions, and simplifying expressions.
  • Don't forget to factor denominators completely before starting.
  • Use methods like butterfly method for simpler problems.

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