Rational Expressions: Homework Overview and Study Guide

Rational Expressions

Introduction to Rational Expressions

Rational expressions are a fundamental topic in College Algebra, involving the manipulation and simplification of fractions whose numerators and denominators are polynomials. Mastery of rational expressions is essential for solving equations, analyzing functions, and understanding more advanced algebraic concepts.

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

• Domain: The domain of a rational expression consists of all real numbers except those that make the denominator zero.

• Key Operations: Simplifying, multiplying, dividing, adding, and subtracting rational expressions.

Simplifying Rational Expressions

To simplify a rational expression, factor both the numerator and denominator and then reduce by canceling common factors.

• Step 1: Factor the numerator and denominator completely.

• Step 2: Identify and cancel any common factors.

• Step 3: State the restrictions on the variable (values that make the denominator zero).

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

• Factor numerator: $x^2 - 9 = (x - 3)(x + 3)$

• Factor denominator: $x^2 - 6x + 9 = (x - 3)^2$

• Cancel $(x - 3)$: $\frac{(x - 3)(x + 3)}{(x - 3)(x - 3)} = \frac{x + 3}{x - 3}$, $x \neq 3$

Multiplying and Dividing Rational Expressions

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

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

• Division: Multiply by the reciprocal of the divisor.

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

• Factor: $x^2 - 4 = (x - 2)(x + 2)$

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

• Cancel $x$ and $(x + 2)$: $\frac{2}{3(x - 2)}$, $x \neq 0, 2, -2$

Adding and Subtracting Rational Expressions

To add or subtract rational expressions, first find a common denominator.

• Step 1: Factor denominators and determine the least common denominator (LCD).

• Step 2: Rewrite each expression with the LCD.

• Step 3: Combine numerators and simplify.

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

• LCD is $x^2$

• Rewrite: $\frac{x}{x^2} + \frac{2}{x^2} = \frac{x + 2}{x^2}$

Solving Rational Equations

Rational equations are equations that contain rational expressions. To solve them, clear denominators by multiplying both sides by the LCD, then solve the resulting equation.

• Step 1: Identify the LCD of all denominators.

• Step 2: Multiply both sides by the LCD to eliminate denominators.

• Step 3: Solve the resulting equation.

• Step 4: Check for extraneous solutions (values that make any denominator zero).

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

• Subtract 2: $\frac{1}{x} = 1$

• Multiply both sides by $x$: $1 = x$

• Solution: $x = 1$

Key Properties and Restrictions

• Always state restrictions on the variable to avoid division by zero.

• Check for extraneous solutions when solving equations.

Summary Table: Operations with Rational Expressions

Operation	Steps	Example
Simplify	Factor, cancel common factors, state restrictions	$\frac{x^2 - 4}{x^2 - 2x} = \frac{(x + 2)(x - 2)}{x(x - 2)} = \frac{x + 2}{x}$, $x \neq 0, 2$
Multiply	Multiply numerators and denominators, simplify	$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Divide	Multiply by reciprocal, simplify	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
Add/Subtract	Find LCD, rewrite, combine numerators, simplify	$\frac{1}{x} + \frac{2}{x^2} = \frac{x + 2}{x^2}$

Additional info: These notes are based on a homework summary for a College Algebra course, specifically focusing on rational expressions, which are a core topic in Chapter 3 of most College Algebra textbooks.