Lecture Notes: Adding and Subtracting Rational Expressions with Unlike Denominators

Introduction

• Focus on adding and subtracting rational expressions with unlike denominators.
• Importance of finding a common denominator before performing addition or subtraction.

Example 1

• Expression: ( \frac{5}{x} + \frac{3}{x^2} )
  • Multiply first fraction by ( \frac{x}{x} ) to get common denominator ( x^2 ).
  • Combine: ( \frac{5x + 3}{x^2} ).

Example 2

• Expression: ( \frac{x-3}{4} - \frac{x+2}{3} )
  • Least Common Multiple (LCM) of 4 and 3 is 12.
  • Multiply first fraction by 3, second by 4.
  • Distribute and combine: ( \frac{-x - 17}{12} ).
  • Simplify: ( \frac{-(x + 17)}{12} ) or ( \frac{-x + 17}{12} ).

Example 3

• Expression: ( \frac{4}{x-2} + \frac{5}{x+2} )
  • Multiply each fraction by the denominator of the other fraction.
  • Denominator becomes ( x^2 - 4 ) (difference of squares).
  • Combine numerators: ( \frac{9x - 2}{x^2 - 4} ).

Example 4

• Expression: ( \frac{x}{x^2 + 9x + 20} - \frac{4}{x^2 + 7x + 12} )
  • Factor denominators and find common denominator.
  • Combine fractions using common denominator and simplify: ( \frac{x-5}{(x+3)(x+4)(x+5)} ).

Example 5

• Expression: ( \frac{x^2}{x-4} + \frac{7}{4-x} )
  • Recognize that (4-x) can be rewritten as (-(x-4)).
  • Combine: ( \frac{x^2 - 7}{x-4} ).

Example 6

• Expression: ( \frac{5}{x+2} + \frac{2}{x+1} - \frac{3}{x-1} )
  • Multiply each fraction by the other denominators to get a common denominator of ((x+2)(x+1)(x-1)).
  • Simplify and combine terms: ( \frac{4x^2 - 7x - 15}{(x+2)(x+1)(x-1)} ).
  • Factor the numerator for final expression.

Conclusion

• Key steps include identifying the least common denominator, multiplying to get common denominators, distributing terms, and simplifying expressions.
• Factoring and canceling common terms where possible to achieve a simplified result.