Lecture Notes: Multiplying and Dividing Rational Expressions

Overview

This lecture focuses on multiplying and dividing rational expressions, simplifying expressions, and identifying non-permissible values.

Key Points

• Multiplying rational expressions is straightforward: multiply numerators together and denominators together.
• Simplify the expression by canceling out common factors in the numerator and denominator.
• Dividing rational expressions requires multiplying by the reciprocal of the divisor.

Examples

Example 1: Basic Multiplication

• Expression: ( \frac{3}{4} \times \frac{2}{6} )
  • Multiply across: ( \frac{3 \times 2}{4 \times 6} = \frac{6}{24} )
  • Simplify: ( \frac{1}{4} )
  • Alternate method: cancel common factors before multiplying.

Example 2: Multiplying with Variables

• Expression: ( \frac{b}{12} \times \frac{3a}{2b^2} )
  • Cancel common factors: ( b ) in numerator with one ( b ) in ( b^2 )
  • Simplify fractions in terms of coefficients: ( \frac{3}{12} = \frac{1}{4} )
  • Result: ( \frac{a}{8b} )

Example 3: Including Non-Permissible Values

• Expression: ( \frac{6m^3(n+1)}{15m(n+1)} )
  • Cancel ( n+1 ) terms
  • Simplify: ( \frac{4m}{5} )
  • Non-permissible values: ( m \neq 0 ), ( n \neq -1 )

Division of Rational Expressions

• Convert division into multiplication by taking the reciprocal of the divisor.

Example 4: Division

• Expression: ( \frac{12x^2}{15} \div \frac{3x}{2y} )
  • Convert: ( \frac{12x^2}{15} \times \frac{2y}{3x} )
  • Simplify by canceling terms.
  • Result: ( \frac{8xy}{15} )
  • Non-permissible values: ( y \neq 0 ), ( x \neq 0 )

Advanced Examples

Example 5: Factoring and Simplifying

• Expression: ( \frac{x^2 - x - 20}{x+4} \times \frac{x+1}{x^2 - 3x - 10} )
  • Factor terms to simplify
  • Identify non-permissible values by considering initial and flipped denominators.
  • Non-permissible values: ( x \neq -4, 1, -2, 5 )

Recap

• Multiplying rational expressions involves multiplying numerators and denominators.
• Dividing involves multiplying by the reciprocal.
• Always simplify expressions to find simplest form.
• Identify non-permissible values before and after simplification.

Practice

• Apply these concepts to various exercises to reinforce understanding and proficiency.

These notes cover key concepts, examples, and non-permissible values associated with multiplying and dividing rational expressions.