Rational Functions: Multiplication, Division, Addition, and Subtraction of Rational Expressions

Mini-Lecture 6.1: Rational Functions and Multiplying and Dividing Rational Expressions

Introduction to Rational Expressions

A rational expression is a fraction in which the numerator and/or the denominator are polynomials. Rational expressions are fundamental in algebra and appear frequently in mathematical modeling and problem solving.

• Definition: A rational expression is any expression that can be written in the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Examples: $\frac{2x+1}{x^2-4}$, $\frac{5}{x}$, $\frac{x^2-9}{x+3}$

Domain of a Rational Expression

The domain of a rational expression consists of all real numbers except those that make the denominator zero. These excluded values are called restrictions.

• To find the domain: Set the denominator equal to zero and solve for the variable. Exclude these values from the domain.

• Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x=2$.

Simplifying Rational Expressions

Simplifying involves factoring the numerator and denominator and canceling any common factors.

• Steps:

  1. Factor both numerator and denominator completely.

  2. Identify and cancel common factors.

  3. State the domain restrictions.

• Example: $\frac{x^2-9}{x^2-6x+9} = \frac{(x-3)(x+3)}{(x-3)(x-3)} = \frac{x+3}{x-3}$, $x \neq 3$

Multiplying and Dividing Rational Expressions

Multiplication and division of rational expressions follow the same rules as for numerical fractions.

• Multiplication: Multiply numerators together and denominators together, then simplify.

• Division: Multiply by the reciprocal of the divisor.

• Formulas:

  • Multiplication: $\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}$

  • Division: $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}$

• Example: $\frac{2x}{x^2-1} \cdot \frac{x+1}{3x} = \frac{2x(x+1)}{(x^2-1)3x} = \frac{2x(x+1)}{3x(x-1)(x+1)} = \frac{2}{3(x-1)}$, $x \neq 0, 1, -1$

Applications of Rational Functions

Rational functions are used to model real-world situations, such as rates, proportions, and cost analysis.

• Example: The cost per book for printing $x$ books is given by $C(x) = \frac{500 + 2x}{x}$, where $500$is a fixed cost and $2x$ is the variable cost.

• Application: To find the cost per book for 300 books, substitute $x=300$ into the formula.

Mini-Lecture 6.2: Adding and Subtracting Rational Expressions

Adding and Subtracting with Common Denominators

When rational expressions have the same denominator, add or subtract the numerators and keep the denominator unchanged.

• Formula: $\frac{A}{C} + \frac{B}{C} = \frac{A+B}{C}$

• Example: $\frac{2x}{x+1} + \frac{3}{x+1} = \frac{2x+3}{x+1}$

Least Common Denominator (LCD)

The Least Common Denominator is the smallest expression that is a common multiple of the denominators of two or more rational expressions.

• To find the LCD:

  1. Factor each denominator completely.

  2. Include each factor the greatest number of times it occurs in any denominator.

  3. Multiply these factors together to get the LCD.

• Example: For $\frac{1}{x^2}$ and $\frac{1}{x}$, the LCD is $x^2$.

Adding and Subtracting with Unlike Denominators

To add or subtract rational expressions with different denominators, first rewrite each expression with the LCD as the denominator, then add or subtract the numerators.

• Steps:

  1. Find the LCD of the denominators.

  2. Rewrite each expression with the LCD as the denominator.

  3. Add or subtract the numerators.

  4. Simplify the result if possible.

• Example: $\frac{1}{x} + \frac{2}{x+1}$

  • LCD is $x(x+1)$

  • Rewrite: $\frac{x+1}{x(x+1)} + \frac{2x}{x(x+1)} = \frac{x+1+2x}{x(x+1)} = \frac{3x+1}{x(x+1)}$

Summary Table: Operations with Rational Expressions

Operation	Steps	Example
Multiplication	Multiply numerators and denominators, then simplify	$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
Division	Multiply by the reciprocal of the divisor	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$
Addition/Subtraction (common denominator)	Add/subtract numerators, keep denominator	$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$
Addition/Subtraction (unlike denominators)	Find LCD, rewrite each with LCD, then add/subtract	$\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$ (if $b$ and $d$ are relatively prime)

Practice Problems (Selected from Images)

• Find the domain:

  • $g(x) = -\frac{6x + x^2}{5x}$

  • $h(x) = \frac{-9}{x+5}$

  • $f(x) = \frac{x}{2x^2 + x}$

• Simplify:

  • $\frac{x^2-81}{9+x}$

  • $\frac{y^2+5y+6}{y^2+10y+21}$

• Multiply/Divide:

  • $\frac{24xy^2}{x^2-49} \cdot \frac{3x-21}{8x^2y^2}$

  • $\frac{x^2+7x}{x^2+8x}$

• Add/Subtract:

  • $\frac{x^2+5x-6}{x^2+9x+18} + \frac{x^2-1}{x^2+7x+12}$

  • $\frac{x^2-1}{x^3-x}$

• Application:

  • The cost per book for printing $x$ books is $C(x) = \frac{500 + 2x}{x}$. Find $C(300)$ and $C(1000)$.

Additional info:

• Some problems and formulas were inferred from context and standard College Algebra curriculum.

• Practice problems are representative of typical rational expression exercises.