8.3 Add and Subtract Rational Expressions with a Common Denominator

Rational expressions are like fractions, but with algebraic terms instead of numbers. They're crucial for solving complex equations and modeling real-world scenarios. This section focuses on adding and subtracting these expressions when they share a common denominator.

When rational expressions have the same denominator, we can add or subtract them by combining their numerators. This process is similar to working with regular fractions, but with variables involved. We'll learn to simplify these expressions and handle special cases.

Adding and Subtracting Rational Expressions with a Common Denominator

Introduction to Rational Expressions

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• Rational expressions are algebraic fractions that represent the quotient of two polynomials
• Like denominators in rational expressions have the same polynomial expression in the denominator
• Unlike denominators in rational expressions have different polynomial expressions in the denominator

Addition of rational expressions

• Add numerators while keeping the common denominator unchanged when adding rational expressions with identical denominators ($\frac{2}{x} + \frac{3}{x} = \frac{2+3}{x} = \frac{5}{x}$)
• Result maintains the same denominator as the original expressions when adding rational expressions with identical denominators ($\frac{7}{y} + \frac{4}{y} = \frac{11}{y}$)

Simplification of rational sums

• Combine like terms or factor the numerator to simplify the result after adding or subtracting rational expressions ($\frac{2x+3}{y} + \frac{x-1}{y} = \frac{3x+2}{y}$)
• Factor the numerator and denominator to identify and cancel out common factors for further simplification ($\frac{x^2-4}{x+2} - \frac{3x+6}{x+2} = \frac{(x-2)(x+2)}{x+2} - \frac{3(x+2)}{x+2} = \frac{(x-2)(x+2)-3(x+2)}{x+2} = \frac{x^2-3x-10}{x+2}$)
  • Simplify fractions by dividing out common factors in the numerator and denominator ($\frac{6x}{9} - \frac{2x}{9} = \frac{4x}{9} = \frac{4}{9}x$)
• Complex fractions (fractions containing fractions) can be simplified by multiplying both numerator and denominator by the LCD of all denominators within the complex fraction

Denominators as opposites in expressions

• Sum equals 0 when adding rational expressions with opposite denominators due to numerators canceling out ($\frac{3}{x} + \frac{3}{-x} = \frac{3-3}{x} = \frac{0}{x} = 0$)
• Difference equals twice the numerator divided by the positive denominator when subtracting rational expressions with opposite denominators, as subtracting a negative is equivalent to adding a positive ($\frac{2}{y} - \frac{2}{-y} = \frac{2-(-2)}{y} = \frac{4}{y}$)