Rational Expressions

In AP Precalculus, rational expressions are algebraic expressions formed by dividing one polynomial by another.
They are essential for modeling ratios, rates, densities, and situations involving constraints.

Mastery of rational expressions is required for:

• Rational functions and their graphs

• Asymptotic behavior

• Domain restrictions

• Preparation for limits in Calculus

Definition of a Rational Expression

A rational expression is any expression that can be written as:

$$\frac{f(x)}{g(x)}$$​

Where:

• f(x) and g(x) are polynomials

• g(x)≠0

Examples

$$\frac{3x+1}{x-2} , \frac{x^2-4}{x^2+1} , \frac{5}{x}$$​

Domain of Rational Expressions

The domain consists of all real numbers except those that make the denominator zero.

Example

$$\frac{2x}{x^2-9}$$​

Denominator:

$$x^2-9=(x+3)(x-3)$$

Excluded values:

$$x=\pm3$$

Simplifying Rational Expressions

Simplification involves:

1. Factoring numerator and denominator

2. Canceling common factors

Example

$$\frac{x^2-4}{x^2+2x}$$​

Factor:

$$\frac{(x+2)(x-2)}{x(x+2)}$$​

Simplify:

$$\frac{x-2}{x}$$ , $$x\neq0,-2$$

Note: Canceling does not remove domain restrictions.

Multiplying Rational Expressions

Steps

1. Factor all polynomials

2. Cancel common factors

3. Multiply remaining factors

Example

$$\frac{3x}{x-1}\cdot\frac{x^2-1}{2x}=\frac{3(x+1)}{2}$$ , $$x\neq0,1$$

Dividing Rational Expressions

Dividing is equivalent to multiplying by the reciprocal.

Example

$$\frac{4x}{x+3}\div\frac{2x}{x-1}=\frac{4x}{x+3}\cdot\frac{x-1}{2x}=\frac{2(x-1)}{x+3}$$​

Adding and Subtracting Rational Expressions

Steps

1. Find the least common denominator (LCD)

2. Rewrite expressions with the LCD

3. Combine numerators

4. Simplify

Example

$$\frac{1}{x}+\frac{2}{x+1}$$​

LCD:

$$x(x+1)$$​

then

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

Complex Rational Expressions

A complex rational expression has rational expressions in the numerator or denominator.

Example

$$\frac{\frac{1}{x}+2}{\frac{3}{x}}$$​

Multiply numerator and denominator by x:

$$\frac{1+2x}{3}$$​

Rational Expressions and Zeros

• Zeros occur when the numerator equals zero

• Denominator must not be zero

Example

$$\frac{x-4}{x^2-1}=0$$ then $$\frac{x-4}{(x+1)(x-1)}=0$$ so $$x=4 , x\neq\pm1$$​

Summary

• Rational expressions are ratios of polynomials

• Domain restrictions arise from denominators

• Simplification requires factoring

• All four operations follow structured rules

• Rational expressions are foundational for rational functions and calculus