8.5 Simplify Complex Rational Expressions

Complex rational expressions can be tricky, but they're just fractions within fractions. We'll learn two main ways to simplify them: rewriting as division problems and multiplying by the LCD. These methods help turn complicated expressions into simpler ones.

Simplifying complex rationals is all about breaking them down into manageable parts. We'll use factoring, common denominators, and basic fraction rules to transform these expressions. With practice, you'll be able to tackle even the most intimidating rational expressions.

Simplifying Complex Rational Expressions

Rewriting as division problem

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• Recognize a complex rational expression contains one or more rational expressions in numerator, denominator, or both ($\frac{x+1}{x-1} \div \frac{x+2}{x+3}$)
• Rewrite complex rational expression as division problem by placing numerator rational expression over denominator rational expression ($\frac{\frac{x+1}{x-1}}{\frac{x+2}{x+3}}$)
• Simplify resulting division problem by dividing numerator by denominator, treating each rational expression as single term
• Factor out common factors in numerator and denominator before dividing if possible ($\frac{x+1}{x-1} \cdot \frac{x+3}{x+2}$)
• Simplify resulting rational expression if needed ($\frac{(x+1)(x+3)}{(x-1)(x+2)}$)
• Consider domain restrictions when simplifying to ensure the final expression is valid for all input values

Multiplying by LCD

• Find least common denominator (LCD) of all rational expressions in numerator and denominator
  • Factor each denominator into prime factors ($x-1 = (x-1), x+3 = (x+3)$)
  • LCD is product of all prime factors, using highest power of each factor that appears in any denominators ($(x-1)(x+3)$)
• Multiply both numerator and denominator of complex rational expression by LCD ($\frac{x+1}{x-1} \cdot \frac{x+3}{x+3} \div \frac{x+2}{x+3} \cdot \frac{x-1}{x-1}$)
  • Eliminates all denominators within numerator and denominator, leaving only polynomial expressions ($\frac{(x+1)(x+3)}{(x-1)(x+3)} \div \frac{(x+2)(x-1)}{(x+3)(x-1)}$)
• Simplify resulting numerator and denominator by combining like terms and factoring if possible ($(x+1) \div (x+2)$)
• Divide simplified numerator by simplified denominator to obtain final simplified rational expression ($\frac{x+1}{x+2}$)

Order of operations in simplification

• Simplify numerator and denominator separately by applying order of operations (PEMDAS)
  • Parentheses: Simplify expressions within parentheses ($\frac{(2x+1)(x-3)}{(x+4)(x-3)} \div \frac{(x+5)(x-3)}{(x+4)(x-3)}$)
  • Exponents: Evaluate all exponents ($\frac{2x^2-5x-3}{x^2+x-12} \div \frac{x^2+2x-15}{x^2+x-12}$)
  • Multiplication and Division: Perform these operations from left to right
  • Addition and Subtraction: Perform these operations from left to right
• Factor out common factors in simplified numerator and denominator if possible ($\frac{(2x-3)(x-1)}{(x+4)(x-3)} \div \frac{(x+5)(x-3)}{(x+4)(x-3)}$)
• Divide simplified numerator by simplified denominator to obtain final simplified rational expression
  • Cancel out common factors in numerator and denominator before dividing ($\frac{2x-3}{x+5}$)

Simplification Techniques and Algebraic Manipulation

• Use reciprocal method to simplify complex fractions by inverting the denominator and multiplying
• Apply algebraic fraction principles to break down complex expressions into simpler components
• Utilize factoring and polynomial division to simplify numerators and denominators
• Employ cross-multiplication for comparing or combining rational expressions
• Recognize and simplify nested fractions within complex rational expressions