Rational Expressions: Domains, Operations, and Complex Forms

Rational Expressions

Introduction

Rational expressions are quotients of polynomials and are a fundamental topic in algebra and precalculus. Understanding their domains, how to simplify, and how to perform operations on them is essential for success in higher mathematics.

Domain of Rational Expressions

The domain of a rational expression is the set of all real numbers for which the expression is defined. Division by zero is undefined, so any value that makes the denominator zero must be excluded from the domain.

• Key Point: To find the domain, set the denominator equal to zero and solve for the variable. Exclude these solutions from the domain.

• Example: Find the domain of $\frac{2}{x-3}$.

Solution:

• Set $x-3=0$

• $x=3$

• The denominator is zero when $x=3$, so the domain is all real numbers except $x=3$.

Simplifying, Multiplying, and Dividing Rational Expressions

To simplify rational expressions, use the properties of fractions and factorization. Multiplication and division follow the same rules as with numerical fractions.

• Key Point: Cancel common factors in the numerator and denominator.

• Multiplication: Multiply numerators and denominators, then simplify.

• Division: Multiply by the reciprocal of the divisor.

Formula:

• $\frac{a}{b} \cdot \frac{c}{c} = \frac{a \cdot c}{b \cdot c} = \frac{a}{b}$

Example: Simplify $\frac{9x^2 + 6x - 3}{12x^2 - 12}$.

• Factor numerator and denominator:

• $9x^2 + 6x - 3 = 3(3x^2 + 2x - 1)$

• $12x^2 - 12 = 12(x^2 - 1) = 12(x+1)(x-1)$

• Simplify:

• $\frac{3(3x^2 + 2x - 1)}{12(x^2 - 1)} = \frac{3(x+1)(3x-1)}{12(x+1)(x-1)} = \frac{3x-1}{4(x-1)}$

Adding and Subtracting Rational Expressions

To add or subtract rational expressions, they must have a common denominator. If denominators differ, find the least common denominator (LCD) by factoring each denominator and using each factor the greatest number of times it occurs in any denominator.

• Key Point: After finding the LCD, rewrite each expression with the LCD as the denominator, then add or subtract the numerators.

Example: Add $\frac{x^2 - 4x + 4}{2x^2 - 3x + 1} + \frac{x + 4}{2x - 2}$.

• Factor denominators:

• $2x^2 - 3x + 1 = (2x - 1)(x - 1)$

• $2x - 2 = 2(x - 1)$

• LCD is $2(2x - 1)(x - 1)$

• Rewrite each fraction with the LCD:

• $\frac{x^2 - 4x + 4}{(2x - 1)(x - 1)} \cdot \frac{2}{2} = \frac{2x^2 - 8x + 8}{2(2x - 1)(x - 1)}$

• $\frac{x + 4}{2(x - 1)} \cdot \frac{2x - 1}{2x - 1} = \frac{2x^2 + 7x - 4}{2(2x - 1)(x - 1)}$

• Add numerators:

• $\frac{2x^2 - 8x + 8 + 2x^2 + 7x - 4}{2(2x - 1)(x - 1)} = \frac{4x^2 - x + 4}{2(2x - 1)(x - 1)}$

Complex Rational Expressions

A complex rational expression is a rational expression whose numerator, denominator, or both contain rational expressions themselves.

• Key Point: Simplify complex rational expressions by either multiplying numerator and denominator by the LCD of all denominators within the expression, or by combining terms and dividing as with simple fractions.

• Method 1: Find the LCD of all denominators within the complex rational expression. Multiply numerator and denominator by this LCD.

• Method 2: Combine terms in the numerator and denominator, then divide by multiplying by the reciprocal of the denominator.

Example:

• Simplify $\frac{\frac{a}{b} + \frac{a}{h}}{\frac{b}{a} + \frac{b}{h}}$

• LCD of all denominators is $a h$

• Multiply numerator and denominator by $a h$:

• Numerator: $a h \left( \frac{a}{b} + \frac{a}{h} \right) = a h \cdot \frac{a}{b} + a h \cdot \frac{a}{h} = \frac{a^2 h}{b} + a^2$

• Denominator: $a h \left( \frac{b}{a} + \frac{b}{h} \right) = a h \cdot \frac{b}{a} + a h \cdot \frac{b}{h} = b h + a b$

• Final simplified form: $\frac{a^2 h + a^2 b}{b h + a b}$

Summary Table: Operations on Rational Expressions

Operation	Steps	Example
Domain	Set denominator to zero, exclude solutions	$\frac{2}{x-3}$, domain: $x \neq 3$
Simplify	Factor, cancel common factors	$\frac{9x^2 + 6x - 3}{12x^2 - 12} = \frac{3x-1}{4(x-1)}$
Multiply/Divide	Multiply numerators/denominators, divide by reciprocal	$\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$
Add/Subtract	Find LCD, rewrite, combine numerators	$\frac{x^2 - 4x + 4}{2x^2 - 3x + 1} + \frac{x + 4}{2x - 2}$
Complex	Multiply by LCD or combine then divide	$\frac{\frac{a}{b} + \frac{a}{h}}{\frac{b}{a} + \frac{b}{h}}$

Additional info: These notes expand on the provided slides with full definitions, step-by-step examples, and a summary table for clarity and exam preparation.