7.3 Simplify Complex Rational Expressions

Complex rational expressions can be tricky, but they're just fractions with fractions inside. We'll learn two ways to simplify them: rewriting as division problems and using the LCD method. Both techniques help us break down these expressions into simpler forms.

Choosing between the division and LCD methods depends on the problem. The division method works well for easily factorable expressions, while the LCD method shines when dealing with different denominators. Practice will help you decide which approach to use.

Simplifying Complex Rational Expressions

Rewriting as division problems

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• Complex rational expressions contain fractions within the numerator, denominator, or both ($\frac{2}{x+1} \div \frac{3}{x-2}$)
• Rewrite the expression as a division problem by placing the numerator followed by $\div$ and the denominator ($\frac{2}{x+1} \div \frac{3}{x-2}$)
• Factor the numerator and denominator completely ($\frac{2}{x+1} \div \frac{3}{x-2}$)
• Identify and cancel out common factors in the numerator and denominator ($\frac{2}{x+1} \div \frac{3}{x-2} = \frac{2(x-2)}{(x+1)3}$)
• Multiply any remaining factors in the numerator and denominator separately ($\frac{2(x-2)}{(x+1)3} = \frac{2x-4}{3x+3}$)
• Write the simplified result as a single fraction ($\frac{2x-4}{3x+3}$)

Simplification using LCD method

• Find the least common denominator (LCD) of all denominators in the expression
  • Factor each denominator completely ($\frac{2}{x+1} + \frac{3}{x-2} \rightarrow$ LCD: $(x+1)(x-2)$)
  • The LCD is the product of all unique factors, using the highest power of each
• Multiply both the numerator and denominator of the complex rational expression by the LCD ($\frac{2}{x+1} + \frac{3}{x-2} = \frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x-2)(x+1)}$)
• Simplify the numerator and denominator by distributing and combining like terms ($\frac{2x-4}{(x+1)(x-2)} + \frac{3x+3}{(x-2)(x+1)}$)
• Factor the resulting numerator and denominator completely ($\frac{2x-4}{(x+1)(x-2)} + \frac{3x+3}{(x-2)(x+1)}$)
• Cancel out common factors ($\frac{2x-4}{(x+1)(x-2)} + \frac{3x+3}{(x-2)(x+1)} = \frac{2x-4+3x+3}{(x+1)(x-2)}$)
• Write the simplified result as a single fraction ($\frac{5x-1}{(x+1)(x-2)}$)

Division vs LCD method efficiency

• The division method is often more efficient when:
  • Numerator and denominator can be factored easily ($\frac{x^2-4}{x+2} \div \frac{x+1}{x-2}$)
  • Common factors can be canceled out ($\frac{(x-2)(x+2)}{x+2} \div \frac{x+1}{x-2} = \frac{x-2}{1}$)
• The LCD method is often more efficient when:
  • Denominators of rational expressions within the complex rational expression have different factors ($\frac{1}{x+1} + \frac{2}{x-3}$)
  • Numerator and denominator cannot be factored easily ($\frac{2x+1}{x^2-4} + \frac{3x-2}{x^2+5x+6}$)
  • No common factors to cancel out ($\frac{2x+1}{x^2-4} + \frac{3x-2}{x^2+5x+6}$)
• In some cases, both methods may be equally efficient
  • Choose the method that you find easier to apply consistently
  • With practice, develop a sense of which method is more suitable for a given problem ($\frac{x+1}{x-2} \div \frac{x-3}{x+4}$ can be solved efficiently using either method)

Understanding Rational Expressions

• Rational expressions are algebraic fractions that represent the quotient of two polynomials
• The numerator and denominator of a rational expression are polynomials
• The reciprocal of a rational expression is formed by flipping the numerator and denominator
• Cross multiplication is a technique used to solve equations involving rational expressions