1.6 Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. They're key for modeling complex relationships in math and science. Simplifying and operating on these expressions involves factoring, canceling common terms, and finding common denominators.

Adding, subtracting, multiplying, and dividing rational expressions requires specific techniques. Solving equations with rational expressions often involves clearing fractions by multiplying both sides by the least common denominator. It's crucial to check solutions and identify restrictions on variables.

Simplifying and Operating on Rational Expressions

Simplification of rational expressions

• Factor numerator and denominator completely
  • Identify greatest common factor (GCF) of terms in numerator and denominator ($x^2 + 2x$, GCF is $x$)
  • Factor out GCF from numerator and denominator ($x(x + 2)$)
  • Factor remaining expressions in numerator and denominator ($x^2 - 4 = (x + 2)(x - 2)$)
• Cancel common factors in numerator and denominator
  • Identify matching factors in numerator and denominator ($(x + 2)$ in both)
  • Divide out (cancel) matching factors, leaving remaining factors in numerator and denominator ($\frac{x}{x - 2}$)
• Combine remaining factors in numerator and denominator
  • Multiply remaining factors in numerator to form new numerator ($x$)
  • Multiply remaining factors in denominator to form new denominator ($x - 2$)

Operations with rational expressions

• Multiplication of rational expressions
  • Factor numerators and denominators of rational expressions ($\frac{x + 1}{x - 3} \cdot \frac{x - 2}{x + 4}$)
  • Multiply numerators together to form new numerator ($(x + 1)(x - 2)$)
  • Multiply denominators together to form new denominator ($(x - 3)(x + 4)$)
  • Simplify resulting rational expression by canceling common factors ($\frac{x^2 - x - 2}{x^2 + x - 12}$)
• Division of rational expressions
  • Rewrite division as multiplication by reciprocal of divisor ($\frac{x + 2}{x - 1} \div \frac{x + 3}{x - 4} = \frac{x + 2}{x - 1} \cdot \frac{x - 4}{x + 3}$)
    • Reciprocal: flip numerator and denominator of divisor ($\frac{x + 3}{x - 4} \rightarrow \frac{x - 4}{x + 3}$)
  • Multiply rational expressions using multiplication rules (see above)
  • Simplify resulting rational expression by canceling common factors ($\frac{(x + 2)(x - 4)}{(x - 1)(x + 3)}$)

Addition and subtraction of rationals

• Like denominators
  • Add or subtract numerators while keeping denominator the same ($\frac{2}{x} + \frac{3}{x} = \frac{2 + 3}{x} = \frac{5}{x}$)
  • Simplify resulting numerator and denominator if possible
• Unlike denominators
  • Find least common denominator (LCD) of rational expressions
    • LCD is the least common multiple (LCM) of individual denominators ($\frac{1}{x - 1} - \frac{2}{x + 1}$, LCD is $(x - 1)(x + 1)$)
  • Rewrite each rational expression as an equivalent expression with LCD
    • Multiply numerator and denominator by factor needed to create LCD ($\frac{1}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{2}{x + 1} \cdot \frac{x - 1}{x - 1}$)
  • Add or subtract numerators of equivalent expressions while keeping LCD as denominator ($\frac{x + 1}{(x - 1)(x + 1)} - \frac{2(x - 1)}{(x - 1)(x + 1)} = \frac{x + 1 - 2(x - 1)}{(x - 1)(x + 1)}$)
  • Simplify resulting numerator and denominator if possible ($\frac{-x + 3}{(x - 1)(x + 1)}$)

Complex rational expressions

• Simplify numerator and denominator separately using rational expression simplification rules
  • Factor and cancel common factors in numerator and denominator ($\frac{\frac{x + 2}{x - 1}}{\frac{x + 3}{x - 4}}$, simplify to $\frac{x + 2}{x + 3} \cdot \frac{x - 4}{x - 1}$)
• Perform division of simplified rational expressions
  • Rewrite as multiplication by reciprocal and simplify ($\frac{(x + 2)(x - 4)}{(x + 3)(x - 1)}$)

Understanding Rational Expressions

• A rational expression is a fraction (ratio) of two polynomials
• The numerator and denominator are algebraic expressions containing variables
• Rational expressions are a type of algebraic expression used to represent complex relationships
• Equations involving rational expressions can model real-world scenarios and require specific solving techniques

Evaluating and Solving Rational Expressions

Equations with rational expressions

• Clear fractions by multiplying both sides of the equation by the LCD of all rational expressions
  • Identify LCD of all rational expressions in the equation ($\frac{1}{x - 2} + \frac{3}{x + 1} = \frac{2}{x - 2}$, LCD is $(x - 2)(x + 1)$)
  • Multiply both sides of the equation by the LCD ($(x - 2)(x + 1) \cdot (\frac{1}{x - 2} + \frac{3}{x + 1}) = (x - 2)(x + 1) \cdot \frac{2}{x - 2}$)
• Simplify each side of the equation by distributing and combining like terms ($x + 1 + 3(x - 2) = 2(x + 1)$)
• Solve resulting equation using appropriate method (e.g., factoring, quadratic formula)
  • Subtract 2x and 2 from both sides ($x + 1 + 3x - 6 = 2x + 2 \rightarrow 4x - 5 = 2x + 2$)
  • Subtract 2x from both sides ($2x - 5 = 2$)
  • Add 5 to both sides ($2x = 7$)
  • Divide both sides by 2 ($x = \frac{7}{2}$)
• Check potential solutions by substituting them back into the original equation
  • Verify that each solution satisfies the equation ($\frac{1}{\frac{7}{2} - 2} + \frac{3}{\frac{7}{2} + 1} = \frac{2}{\frac{7}{2} - 2} \rightarrow \frac{1}{\frac{3}{2}} + \frac{3}{\frac{9}{2}} = \frac{2}{\frac{3}{2}} \rightarrow \frac{2}{3} + \frac{2}{3} = \frac{4}{3}$, which is true)
• Identify restrictions on variables that would make any denominator equal to zero
  • These values are not part of the solution set of the equation ($x \neq 2$ and $x \neq -1$ in the original equation)