Subtract Rational Expressions

Algebra-2

1. Fundamental Concepts

Definition: Rational expressions are fractions where the numerator and denominator are polynomials.
Subtraction: Subtracting rational expressions involves finding a common denominator and then subtracting the numerators.
Common Denominator: The least common denominator (LCD) is used to combine rational expressions.

2. Key Concepts

Basic Rule: $$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$

Degree Preservation: The degree of the resulting polynomial in the numerator does not necessarily match the degrees of the original numerators.

Application: Used in solving real-world problems involving rates, proportions, and more.

3. Examples

Example 1 (Basic)

Problem: Simplify $$\frac{3x + 2}{x - 1} - \frac{x - 4}{x - 1}$$

Step-by-Step Solution:

Since the denominators are the same, subtract the numerators directly: $$\frac{(3x + 2) - (x - 4)}{x - 1}$$
Simplify the numerator: $$\frac{3x + 2 - x + 4}{x - 1} = \frac{2x + 6}{x - 1}$$

Validation: Substitute \(x = 2\) → Original: \(\frac{8}{1} - \frac{-2}{1} = 10\); Simplified: \(\frac{10}{1} = 10\) ✓

Example 2 (Intermediate)

Problem: $$\frac{2y^2 + 3y}{y^2 - 4} - \frac{y^2 - 2y}{y^2 - 4}$$

Step-by-Step Solution:

Since the denominators are the same, subtract the numerators directly: $$\frac{(2y^2 + 3y) - (y^2 - 2y)}{y^2 - 4}$$
Simplify the numerator: $$\frac{2y^2 + 3y - y^2 + 2y}{y^2 - 4} = \frac{y^2 + 5y}{y^2 - 4}$$

Validation: Substitute \(y = 1\) → Original: \(\frac{5}{-3} - \frac{-1}{-3} = \frac{4}{-3}\); Simplified: \(\frac{6}{-3} = -2\) ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding for different terms and factors.
Error-Proofing: Double-check the LCD and ensure all terms are correctly aligned before performing operations.
Concept Reinforcement: Practice with various types of rational expressions to build confidence and proficiency.