7.1 Multiply and Divide Rational Expressions

Rational expressions are fractions with algebraic terms. They're key in algebra, helping us solve complex problems and model real-world scenarios. Understanding how to work with them is crucial for tackling more advanced math concepts.

In this section, we'll learn to simplify, multiply, and divide rational expressions. We'll also explore rational functions and their operations. These skills are essential for solving equations and analyzing relationships between variables in various fields.

Rational Expressions

Undefined values in rational expressions

• Rational expression undefined when denominator equals zero
• Find undefined values by setting denominator equal to zero and solving for variable
  • $\frac{x+1}{x-2}$ undefined when $x-2=0$
    • Solve $x-2=0$
    • $x=2$
    • Rational expression undefined when $x=2$
• The domain of a rational expression excludes values that make the denominator zero

Simplification of complex rationals

• Factor numerator and denominator completely
• Cancel out common factors in numerator and denominator
• Multiply remaining factors in numerator and denominator separately
• Simplify $\frac{x^2-4}{x^2+3x-10}$
  • Factor numerator: $x^2-4=(x+2)(x-2)$
  • Factor denominator: $x^2+3x-10=(x+5)(x-2)$
  • Cancel out common factor $(x-2)$
  • Simplified expression: $\frac{x+2}{x+5}$

Multiplication of rational expressions

• Factor numerators and denominators of rational expressions
• Multiply numerators together and denominators together
• Cancel out common factors in resulting numerator and denominator
• Multiply $\frac{2x+6}{x-1}$ and $\frac{x+4}{3x+9}$
  • $\frac{2x+6}{x-1} \cdot \frac{x+4}{3x+9} = \frac{(2x+6)(x+4)}{(x-1)(3x+9)}$
  • Multiply numerators: $(2x+6)(x+4)=2x^2+14x+24$
  • Multiply denominators: $(x-1)(3x+9)=3x^2+6x-9$
  • Resulting product: $\frac{2x^2+14x+24}{3x^2+6x-9}$

Division of rational expressions

• Rewrite division problem as multiplication problem by multiplying first rational expression by reciprocal of second
  • Reciprocal of rational expression found by swapping numerator and denominator
• Factor numerators and denominators, then multiply and cancel out common factors
• Divide $\frac{3x^2-12}{2x+4}$ by $\frac{x+1}{x-2}$
  • $\frac{3x^2-12}{2x+4} \div \frac{x+1}{x-2} = \frac{3x^2-12}{2x+4} \cdot \frac{x-2}{x+1}$
  • Factor numerators and denominators
    • $3x^2-12=3(x^2-4)=3(x+2)(x-2)$
    • $2x+4=2(x+2)$
  • Cancel out common factors and multiply remaining factors
    • $\frac{3x^2-12}{2x+4} \cdot \frac{x-2}{x+1} = \frac{3(x+2)(x-2)}{2(x+2)} \cdot \frac{x-2}{x+1} = \frac{3(x-2)}{2} \cdot \frac{x-2}{x+1} = \frac{3(x-2)^2}{2(x+1)}$

Rational function operations

• Rational functions are functions that can be written as ratio of two polynomials
• To multiply or divide rational functions, follow same steps as multiplying or dividing rational expressions
• Be cautious of undefined values when working with rational functions
  • Avoid inputting values that result in denominator being zero
• Given rational functions $f(x)=\frac{2x-1}{x+3}$ and $g(x)=\frac{x^2+4x+3}{x-1}$, find $f(x) \cdot g(x)$
  • $f(x) \cdot g(x) = \frac{2x-1}{x+3} \cdot \frac{x^2+4x+3}{x-1}$
  • Factor numerators and denominators
    • $2x-1$ and $x+3$ cannot be factored further
    • $x^2+4x+3=(x+1)(x+3)$
  • Cancel out common factors and multiply remaining factors
    • $\frac{2x-1}{x+3} \cdot \frac{x^2+4x+3}{x-1} = \frac{2x-1}{x+3} \cdot \frac{(x+1)(x+3)}{x-1} = (2x-1)(x+1)$
  • Resulting product: $f(x) \cdot g(x) = (2x-1)(x+1)$

Components of Rational Expressions

• Rational expressions are fractions containing algebraic expressions
• Numerator and denominator are polynomials, which are sums of terms with variables and exponents
• Variables in rational expressions represent unknown quantities
• Exponents indicate how many times a base is multiplied by itself

Applying Rational Expressions