Simplifying Rational Expressions

Introduction to Simplifying Rational Expressions

• Focus on simplifying rational expressions.
• Key techniques: factoring, cancelling, and simplifying.

Example 1: Simplifying 35x⁵ / 49x²

• Reduce coefficients: 35 and 49 by 7.
  • 35 ÷ 7 = 5
  • 49 ÷ 7 = 7
• Subtract exponents of common base: 5 - 2 = 3.
• Final Simplified Expression: 5x³ / 7.

Steps for Simplification

• Start with Factorization:
  • 35 = 7 x 5
  • x⁵ = x x x x x
  • 49 = 7 x 7
  • x² = x x
• Cancel Common Factors:
  • Cancel 7 and two x's.
• Result: 5x³ / 7.

Example 2: Simplifying (4x² + 8x) / (3x + 6)

• Factor GCF:
  • Numerator GCF: 4x
    • (4x² / 4x = x), (8x / 4x = 2)
  • Denominator GCF: 3
    • (3x / 3 = x), (6 / 3 = 2)
• Cancel Common Terms: x + 2
• Final Expression: 4x / 3.

Example 3: Simplifying (x² - 16) / (x² + 9x + 20)

• Use Difference of Squares: x² - 16 = (x + 4)(x - 4)
• Factor Trinomial: x² + 9x + 20
  • Find numbers multiplying to 20 and adding to 9: 4 and 5
  • Expression: (x + 4)(x + 5)
• Cancel Common Factor: x + 4
• Final Expression: (x - 4) / (x + 5).

Special Case Example

• Expression: (5 - x) / (x - 5)
• Factor out -1: from numerator
  • Result: -1(x - 5)
• Cancel Common Terms: x - 5
• Final Expression: -1.

Practice Problems

1. Problem: 72x⁸y⁷ / 64x⁵y⁴
   • Factor & cancel common factors.
   • Result: 9x³y³ / 8.
2. Problem: (5x² - 15x) / (8x - 24)
   • Factor out GCFs: 5x and 8.
   • Cancel x - 3.
   • Result: 5x / 8.
3. Problem: (42 - 6x) / (3x - 21)
   • Factor out GCFs: 6 and 3.
   • Factor out -1 and reverse terms.
   • Cancel x - 7.
   • Result: -2.
4. Problem: (x² - 8x + 15) / (2x² - 18)
   • Factor trinomial and use difference of squares technique.
   • Result: (x - 5) / (2(x + 3)).
5. Problem: (2x² - 5x - 3) / (4x² - 1)
   • Factor using grouping and difference of squares.
   • Cancel 2x + 1.
   • Result: (x - 3) / (2x - 1).

Conclusion

• Techniques: Factor, take out the GCF, and cancel.
• Practice with examples to master simplification of rational expressions.