1.6 Rational Expressions

Rational expressions are like mathematical fractions on steroids. They involve polynomials in both the top and bottom parts. Simplifying these expressions is crucial for solving complex math problems and understanding their behavior.

Operations with rational expressions follow similar rules to regular fractions, but with a twist. Addition, subtraction, multiplication, and division all require special techniques. Solving equations with rationals and analyzing complex rational expressions are advanced skills that build on these fundamentals.

Simplifying and Operating on Rational Expressions

Simplification of rational expressions

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• Factor numerator and denominator completely
  • Find greatest common factor (GCF) of terms in numerator and denominator
  • Factor out GCF from both numerator and denominator
  • Factor remaining polynomial expressions using techniques like:
    • Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
    • Perfect square trinomials: $a^2 + 2ab + b^2 = (a+b)^2$ and $a^2 - 2ab + b^2 = (a-b)^2$
    • Grouping: $ac + ad + bc + bd = (a+b)(c+d)$
• Cancel common factors in numerator and denominator
  • Find factors appearing in both numerator and denominator
  • Divide out common factors to simplify rational expression

Operations with rational expressions

• Multiplication of rational expressions
  • Multiply numerators together for new numerator
  • Multiply denominators together for new denominator
  • Simplify resulting rational expression by factoring and canceling common factors
• Division of rational expressions
  • Rewrite division as multiplication by reciprocal of divisor
    • Reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$
  • Multiply numerators together for new numerator
  • Multiply denominators together for new denominator
  • Simplify resulting rational expression by factoring and canceling common factors

Addition and subtraction of rationals

• Adding and subtracting rational expressions with like denominators
  • Keep common denominator
  • Add or subtract numerators
  • Simplify resulting rational expression by factoring and canceling common factors
• Adding and subtracting rational expressions with unlike denominators
  • Find least common denominator (LCD) of rational expressions
    • LCD is least common multiple (LCM) of denominators
  • Rewrite each rational expression with LCD as denominator
    • Multiply numerator and denominator of each expression by factor needed to obtain LCD
  • Add or subtract numerators of equivalent expressions
  • Simplify resulting rational expression by factoring and canceling common factors

Solving Equations and Analyzing Complex Rational Expressions

Equations with rational expressions

• Clear denominators by multiplying both sides of equation by LCD of all rational expressions
• Simplify resulting equation by distributing and combining like terms
• Solve simplified equation using appropriate techniques like:
  • Factoring and applying zero-product property
  • Using quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ for equations in form $ax^2 + bx + c = 0$
• Check solutions by substituting back into original equation
  • Reject solutions that result in denominator equal to zero (extraneous)

Analysis of complex rationals

• Identify main fraction bar and consider expressions above and below it as separate units
• Simplify numerator and denominator separately by factoring and canceling common factors
• If numerator or denominator contains fractions, use techniques for adding, subtracting, multiplying, or dividing rational expressions to simplify
• Combine simplified numerator and denominator to form simplified complex rational expression
• If necessary, repeat process until expression cannot be simplified further

Key Concepts in Rational Expressions

• Domain: The set of all possible input values for which a rational expression is defined
• Asymptote: A line that the graph of a rational function approaches but never crosses
• Polynomial: An expression consisting of variables and coefficients involving only addition, subtraction, and multiplication operations
• Rational expressions are fractions where both numerator and denominator are polynomials