Rational Expressions and Their Simplification in Precalculus

Rational Expressions

Definition and Structure

A rational expression is a fraction in which the numerator and denominator are both polynomials. The general form is $\frac{a}{b}$, where a and b are polynomials and b is not zero.

• Domain: The set of all real numbers for which the denominator is not zero.

• Example: $\frac{3x+2}{x^2-1}$ is a rational expression.

Simplifying Rational Expressions

Factoring and Reducing

To simplify a rational expression, factor both the numerator and denominator completely, then divide out any common factors. Always specify the domain, noting any restrictions where the denominator is zero.

• Step 1: Factor numerator and denominator.

• Step 2: Identify and divide common factors.

• Step 3: State the domain, excluding values that make the denominator zero.

• Example: Simplify $\frac{3x+3}{x^2-1}$.

Dividing Common Factors

After factoring, cancel out common factors in the numerator and denominator. However, remember that values which make the original denominator zero are still excluded from the domain.

• Example: $\frac{(x+3)(x-1)}{(x+4)(x+4)}$ simplifies to $\frac{x+3}{x+4}$, but $x \neq -4$.

Domain Restrictions

Even after simplification, the domain must exclude any values that made the original denominator zero, even if those factors are cancelled.

• Restriction: Values that make the denominator zero are not in the domain.

• Example: If $x-1$ was cancelled, $x=1$ is still excluded.

Operations with Rational Expressions

Multiplication and Division

To multiply or divide rational expressions, factor all numerators and denominators, multiply across, and then simplify by cancelling common factors.

• Example: $\frac{6x+4}{2x-8} \cdot \frac{x-4}{9x+6}$

• Factor: $6x+4 = 2(x+2)$, $2x-8 = 2(x-4)$, $9x+6 = 3(x+2)$

• Simplify: $\frac{2(x+2)}{2(x-4)} \cdot \frac{x-4}{3(x+2)} = \frac{1}{3}$ (after cancelling common factors)

Addition and Subtraction

To add or subtract rational expressions, find a common denominator, rewrite each fraction with this denominator, then combine numerators and simplify.

• Example: $\frac{4}{3-x} + \frac{2x}{x-3}$

• Note: $x-3$ and $3-x$ are negatives of each other.

• Rewrite: $\frac{4}{3-x} = -\frac{4}{x-3}$

• Combine: $-\frac{4}{x-3} + \frac{2x}{x-3} = \frac{2x-4}{x-3}$

Simplifying Complex Rational Expressions

Methods for Simplification

Complex rational expressions involve fractions within fractions. There are two main methods for simplification:

• Method 1: Simplify numerator and denominator separately, then divide.

• Method 2: Multiply numerator and denominator by the least common denominator (LCD) of all inner fractions to clear denominators.

Example of Complex Rational Expression

Consider $\frac{x(x+h)\frac{1}{x+h} - \frac{x}{x}}{h}$:

• Rewrite numerator: $x(x+h)\frac{1}{x+h} - \frac{x}{x}$

• Simplify: $x - 1$

• Final simplification: $\frac{x-x-h}{xh(x+h)} = \frac{-h}{xh(x+h)} = \frac{-1}{x(x+h)}$

Summary Table: Steps for Simplifying Rational Expressions

Step	Description
1	Factor numerator and denominator completely
2	Identify and cancel common factors
3	State domain restrictions (exclude values making denominator zero)
4	For addition/subtraction, find common denominator
5	For complex expressions, use LCD to clear denominators

Practice Problems

• Simplify $\frac{x^2-9}{x^2-6x+9} \cdot \frac{5x-15}{x+3}$ and state the domain.

• Simplify $\frac{6x+4}{2x-8} \cdot \frac{x-4}{9x+6}$ and state the domain.

• Add $\frac{4}{3-x} + \frac{2x}{x-3}$ and simplify.

Additional info: These notes cover Precalculus Chapter P (Basic Concepts of Algebra) and Chapter 3 (Polynomial and Rational Functions), specifically focusing on rational expressions, their simplification, and domain restrictions.