Rational Expressions: Homework Overview and Study Guide

Rational Expressions

Introduction to Rational Expressions

Rational expressions are algebraic fractions where both the numerator and the denominator are polynomials. Understanding how to manipulate and simplify rational expressions is essential for solving equations and inequalities in College Algebra.

• 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 $Q(x) \neq 0$.

• Key Properties: Rational expressions can be added, subtracted, multiplied, and divided (except by zero).

• Domain: The domain of a rational expression excludes values of $x$ that make the denominator zero.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring polynomials and reducing common factors in the numerator and denominator.

• Step 1: Factor both the numerator and denominator completely.

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

Operations with Rational Expressions

Operations include addition, subtraction, multiplication, and division. Each operation requires finding a common denominator or multiplying/dividing numerators and denominators.

• Addition/Subtraction: Find the least common denominator (LCD), rewrite each expression with the LCD, then add or subtract numerators. Example: $\frac{2}{x} + \frac{3}{x+1}$ LCD is $x(x+1)$ $\frac{2(x+1) + 3x}{x(x+1)} = \frac{2x + 2 + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)}$

• Multiplication: Multiply numerators together and denominators together, then simplify. Example: $\frac{2x}{x^2 - 1} \times \frac{x + 1}{3x}$ $= \frac{2x(x + 1)}{(x^2 - 1)3x}$ Factor $x^2 - 1 = (x - 1)(x + 1)$ $= \frac{2x(x + 1)}{3x(x - 1)(x + 1)}$ Cancel $(x + 1)$ and $x$ $= \frac{2}{3(x - 1)}$

• Division: Multiply by the reciprocal of the divisor. Example: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Solving Equations Involving Rational Expressions

To solve equations with rational expressions, clear denominators by multiplying both sides by the LCD, then solve the resulting polynomial equation.

• Step 1: Identify the LCD of all rational expressions.

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

• Step 3: Solve the resulting equation.

• Step 4: Check solutions against restrictions to avoid extraneous solutions.

• Example: Solve $\frac{2}{x} = \frac{3}{x+1}$ LCD is $x(x+1)$ $2(x+1) = 3x$ $2x + 2 = 3x$ $2 = x$ Check: $x \neq 0, x \neq -1$

Applications of Rational Expressions

Rational expressions are used in various applications such as rates, proportions, and modeling real-world scenarios.

• Example: If a car travels $d$ miles in $t$ hours, its average speed is $\frac{d}{t}$ miles per hour.

• Example: In chemistry, concentration can be expressed as $\frac{\text{amount of solute}}{\text{volume of solution}}$.

Summary Table: Key Concepts in Rational Expressions

Concept	Description	Example
Simplifying	Reduce by factoring and canceling common terms	$\frac{x^2 - 4}{x - 2} = x + 2$ for $x \neq 2$
Adding/Subtracting	Find LCD, combine numerators	$\frac{1}{x} + \frac{2}{x+1} = \frac{x+1 + 2x}{x(x+1)}$
Multiplying	Multiply numerators and denominators, then simplify	$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Dividing	Multiply by reciprocal	$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$
Solving Equations	Clear denominators, solve, check restrictions	$\frac{2}{x} = \frac{3}{x+1}$

Additional info: These notes are based on the homework assignment "P.4 HW - Rational Expressions" for College Algebra, covering the essential skills and concepts needed for mastery of rational expressions.