Rational Expressions: Fresh Take

  • Simplify, multiply, and divide rational expressions.
  • Add and subtract rational expressions, making sure to correctly handle the denominators.

Rational Expressions

The Main Idea

  • Definition:
    • A rational expression is a fraction of polynomials: [latex]\frac{P(x)}{Q(x)}[/latex] where [latex]P(x)[/latex] and [latex]Q(x)[/latex] are polynomials
  • Simplification Process:
    • Factor both numerator and denominator
    • Cancel common factors
  • Key Concept:
    • Only cancel factors, not individual terms

 

Simplify [latex]\dfrac{x - 6}{{x}^{2}-36}[/latex].

Multiplying Rational Expressions

The Main Idea

  • Key Concept:
    • Multiplication of rational expressions follows the same rules as multiplication of fractions
  •  Process:
    • Factor numerators and denominators
    • Multiply numerators together
    • Multiply denominators together
    • Simplify the result
  • Simplification:
    • Cancel common factors between numerator and denominator before multiplying

 

Multiply the rational expressions and show the product in simplest form:

[latex]\dfrac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\cdot \dfrac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[/latex]

Dividing Rational Expressions

The Main Idea

  • Key Concept:
    • Division of rational expressions is equivalent to multiplication by the reciprocal
  • Process:
    • Rewrite as multiplication by reciprocal
    • Factor numerators and denominators
    • Multiply numerators together
    • Multiply denominators together
    • Simplify the result
  • Formula

 

Divide the rational expressions and express the quotient in simplest form:

[latex]\dfrac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\div \dfrac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[/latex]

Adding and Subtracting Rational Expressions

The Main Idea

  • Key Concept:
    • Addition and subtraction of rational expressions follow the same rules as addition and subtraction of fractions
  •  Process:
    • Find the Least Common Denominator (LCD)
    • Rewrite expressions with the LCD
    • Add or subtract the numerators
    • Simplify the result
  • Least Common Denominator (LCD):
    • Smallest multiple that the denominators have in common
    • Found by factoring denominators and multiplying all distinct factors

Add the rational expressions: [latex]\dfrac{2}{x-1} + \dfrac{3}{x+2}[/latex]

Subtract the rational expressions: [latex]\dfrac{3}{x+5}-\dfrac{1}{x - 3}[/latex].

Simplifying Complex Rational Expressions

The Main Idea

  • Key Concept:
    • Every complex rational expression can be simplified to a standard rational expression
  •  Definition:
    • A complex rational expression is a fraction that contains one or more fractions in its numerator, denominator, or both
  • Simplification Process:
    • Combine expressions in the numerator into a single fraction
    • Combine expressions in the denominator into a single fraction
    • Divide the numerator by the denominator
    • Rewrite as multiplication by the reciprocal
    • Multiply and simplify
  • Technique:
    • Use the LCD method to combine fractions within the numerator or denominator

 

Simplify: [latex]\dfrac{\dfrac{x}{y}-\dfrac{y}{x}}{y}[/latex]