Multiplying and Dividing Rational Expressions Lecture

Introduction

• Topic: Multiplying and dividing rational expressions.
• Rational Function: Function with a polynomial over a polynomial (fraction form).
• Rational Expression: Expression on either side of the equal sign, focusing on simplification.
• Goal: Simplify fractions by identifying and canceling common factors while ensuring denominators don't equal zero, which imposes restrictions.

Key Concepts

Rational Functions vs. Rational Expressions

• Rational Function: Can be graphed, involves solving equations.
• Rational Expression: Simplifies expressions, does not involve solving.

Simplification and Restrictions

• Simplified Form: No common factors between numerator and denominator other than ±1.
• Domain Restrictions: Denominator cannot be zero.

Simplifying Rational Expressions

• Common Mistakes:
  • Incorrectly canceling terms in complex fractions.
  • Misunderstanding when simplification is possible.
• Correct Method:
  • Only cancel factors, not terms separated by addition/subtraction.
  • Ensure common factors exist across the entire numerator and denominator.
• Multiplication and Simplification:
  • Factors can be canceled when entirely multiplication (e.g., (3×5)/(13×5), cancel the 5s).
  • Restrictions defined by the denominator.

Steps for Simplifying

1. Identify Restrictions: Determine values where the denominator equals zero.
2. Factor Expressions: Convert into factored form to identify common factors.
3. Cancel Common Factors: Simplify by removing common factors in the numerator and denominator.
4. Consider Restrictions Throughout: List restrictions clearly as part of the answer.

Examples and Practice Problems

Example 1: Simplification Process

• Given two expressions, find common factors.
• Identify restrictions (e.g., x ≠ -1, x ≠ -3).
• Simplify by canceling common factors.

Example 2: Factoring and Restrictions

• Difference of Squares: E.g., x²-16 = (x+4)(x-4).
• Restrictions: Exclude values that cause denominators to be zero.
• Answer Format: Simplified expression combined with restrictions.

Division of Rational Expressions

• Conversion: Division changes to multiplication by reciprocal.
• Steps:
  1. Flip the second fraction.
  2. Factor all expressions.
  3. Simplify and cancel common factors.
  4. List restrictions.

Practice and Homework

• Practice Sheets Available: Factoring practice sheets accessible online.
• Assignments: Practice problems assigned, focusing on methodical simplification and application of restrictions.

Advanced Considerations

• Complex Fractions: Multiple steps in simplification.
• Calculator Use: Encourage mental simplification methods.

Conclusion

• Key Takeaway: Simplifying rational expressions involves careful factoring, cancellation, and mindful tracking of restrictions to ensure valid solutions.
• Next Steps: Assignments on Big Ideas platform, further practice with PDF resources available.