Adding and Subtracting Rational Expressions in Intermediate Algebra

Adding and Subtracting Rational Expressions

Introduction

Rational expressions are fractions in which the numerator and denominator are polynomials. In Intermediate Algebra, understanding how to add and subtract rational expressions is essential for solving more complex algebraic equations. This section covers methods for combining rational expressions with both common and unlike denominators, as well as finding the least common denominator (LCD).

Add or Subtract Rational Expressions with a Common Denominator

When rational expressions share the same denominator, the process of addition or subtraction is straightforward. The numerators are combined, and the denominator remains unchanged.

• Step 1: Add or subtract the numerators while maintaining the common denominator.

• Step 2: Factor the resulting numerator if possible and simplify the fraction.

Formula:

$\frac{P}{Q} + \frac{R}{Q} = \frac{P + R}{Q}$

$\frac{P}{Q} - \frac{R}{Q} = \frac{P - R}{Q}$

Example:

Add $\frac{x + 3}{x + 6} + \frac{x - 4}{x + 6}$

$\frac{x + 3}{x + 6} + \frac{x - 4}{x + 6} = \frac{(x + 3) + (x - 4)}{x + 6} = \frac{2x - 1}{x + 6}$

Example:

Subtract $\frac{a}{a - 6} - \frac{3a - 12}{a - 6}$

$\frac{a}{a - 6} - \frac{3a - 12}{a - 6} = \frac{a - (3a - 12)}{a - 6} = \frac{-2a + 12}{a - 6}$

Example:

Subtract $\frac{3 + 2y}{y^2 - 1} - \frac{y + 3}{y^2 - 1}$

$\frac{3 + 2y}{y^2 - 1} - \frac{y + 3}{y^2 - 1} = \frac{3 + 2y - (y + 3)}{y^2 - 1} = \frac{y + 6}{y^2 - 1}$

Finding the Least Common Denominator (LCD)

The LCD is the smallest expression that can be used as a common denominator for two or more rational expressions. It is essential when adding or subtracting rational expressions with unlike denominators.

• Step 1: Factor each denominator completely.

• Step 2: Identify the LCD of any numerical factors.

• Step 3: For variable factors, use the highest power present in any denominator.

• Step 4: Multiply all factors found in steps 2 and 3 to obtain the LCD.

Example:

• Find the LCD for $\frac{1}{18x^3y}$ and $\frac{5}{27x^2y^3}$ Factor denominators: $18x^3y = 2 \cdot 3^2 \cdot x^3 \cdot y$, $27x^2y^3 = 3^3 \cdot x^2 \cdot y^3$ LCD: $2 \cdot 3^3 \cdot x^3 \cdot y^3 = 54x^3y^3$

• Find the LCD for $\frac{7}{x^2(x + 1)}$ and $\frac{3z}{x(x + 1)^3}$ LCD: $x^2(x + 1)^3$

• Find the LCD for $\frac{4x}{x^2 - x - 12}$ and $\frac{6x^2}{x^2 - 7x + 12}$ Factor denominators: $x^2 - x - 12 = (x - 4)(x + 3)$, $x^2 - 7x + 12 = (x - 3)(x - 4)$ LCD: $(x - 4)(x + 3)(x - 3)$

Add or Subtract Rational Expressions with Unlike Denominators

When denominators are different, the expressions must be rewritten with a common denominator before combining.

• Step 1: Determine the LCD of all denominators.

• Step 2: Rewrite each fraction as an equivalent fraction with the LCD.

• Step 3: Add or subtract the numerators while maintaining the LCD.

• Step 4: Factor and simplify the resulting numerator if possible.

Example:

• $\frac{2}{x^2y} + \frac{5}{3x^3y}$ LCD: $3x^3y$ Rewrite: $\frac{2 \cdot 3x}{3x^3y} + \frac{5}{3x^3y} = \frac{6x + 5}{3x^3y}$

• $\frac{2x - 6}{x - 1} - \frac{4}{1 - x}$ Note: $1 - x = -(x - 1)$ Rewrite: $\frac{2x - 6}{x - 1} + \frac{4}{x - 1} = \frac{2x - 6 + 4}{x - 1} = \frac{2x - 2}{x - 1}$

• $\frac{2x}{x^2 - 9} - \frac{3}{x^2 - 4x + 3}$ Factor denominators: $x^2 - 9 = (x - 3)(x + 3)$, $x^2 - 4x + 3 = (x - 1)(x - 3)$ LCD: $(x - 3)(x + 3)(x - 1)$

Practice Exercises

Try the following exercises to reinforce your understanding:

• $\frac{4}{x^2y - 2} \div \frac{x^2y}{1}$

• $\frac{2}{x - 2} + \frac{x}{x - 2}$

• $\frac{5x + 2}{x^2 + 2x - 8} + \frac{2 - 4x}{x^2 + 2x - 8}$

• $\frac{10}{3x - 3} + \frac{1}{7x - 7}$

• $\frac{-2}{x^2 - 3x} - \frac{1}{x^3 - 3x^2}$

• $\frac{2x}{3x^2 - 13x + 4} + \frac{5}{x^2 - 2x - 8}$

• $\frac{5}{3x - 6} - \frac{x}{x - 2} + 3 + \frac{2x}{5x - 10}$

• $\frac{3}{x + 3} + \frac{5}{x^2 + 6x + 9} - \frac{x}{x^2 - 9}$

Additional info: These exercises are designed to practice the skills of finding the LCD, rewriting rational expressions, and performing addition or subtraction. Always factor denominators and numerators where possible to simplify your answers.