7.2 Add and Subtract Rational Expressions

Rational expressions are fractions with algebraic terms. Adding and subtracting them requires finding common denominators and combining numerators. This process is similar to working with regular fractions, but with variables involved.

Understanding these operations is crucial for solving complex equations and simplifying algebraic expressions. Mastering this skill helps in various math and science applications, from basic algebra to advanced calculus and physics.

Adding and Subtracting Rational Expressions

Addition of rational expressions

• Add rational expressions with common denominators by combining the numerators using addition while keeping the denominator the same
  • $\frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7}$
  • $\frac{4x}{y^2} + \frac{7x}{y^2} = \frac{4x+7x}{y^2} = \frac{11x}{y^2}$
• Simplify the resulting numerator by combining like terms or performing any necessary arithmetic operations, but leave the denominator unchanged
• This process applies to various types of rational expressions, including algebraic expressions and polynomial expressions

Least common denominator identification

• To add or subtract rational expressions with unlike denominators, determine the least common denominator (LCD) first
  • LCD is the least common multiple (LCM) of the individual denominators
  • For $\frac{2}{3}$ and $\frac{5}{6}$, the LCD is 6 (LCM of 3 and 6)
  • For $\frac{x}{2y}$ and $\frac{3}{4y}$, the LCD is 4y (LCM of 2y and 4y)
• Multiply both the numerator and denominator of each rational expression by the factor needed to obtain the LCD
  • To add $\frac{1}{4}$ and $\frac{3}{6}$, multiply the first expression by $\frac{3}{3}$ to get $\frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}$ and the second expression by $\frac{2}{2}$ to get $\frac{3 \cdot 2}{6 \cdot 2} = \frac{6}{12}$

Subtraction with unlike denominators

• After finding the LCD and rewriting the rational expressions with this common denominator, subtract the numerators and keep the LCD as the denominator
  • $\frac{5}{6} - \frac{1}{3} = \frac{10}{12} - \frac{4}{12} = \frac{10-4}{12} = \frac{6}{12} = \frac{1}{2}$
  • $\frac{3x}{4y} - \frac{2x}{3y} = \frac{9x}{12y} - \frac{8x}{12y} = \frac{9x-8x}{12y} = \frac{x}{12y}$
• Simplify the resulting expression by reducing the fraction or performing any necessary arithmetic operations
• Cross multiplication can be used to verify the result of subtraction with unlike denominators

Simplification of complex rationals

• After adding or subtracting rational expressions, simplify the result by factoring the numerator and denominator, if possible
  • Look for common factors in the numerator and denominator and cancel them out
  • $\frac{6x^2+3x}{2x} - \frac{4x^2-2x}{x} = \frac{3x(2x+1)}{2x} - \frac{2x(2x-1)}{x} = \frac{3(2x+1)}{2} - \frac{2(2x-1)}{1} = \frac{6x+3-4x+2}{2} = \frac{2x+5}{2}$
• The simplified result is $\frac{2x+5}{2}$

Techniques for rational functions

• Apply the same techniques used for adding and subtracting rational expressions to rational functions
  • A rational function is a function in the form of a rational expression, where both the numerator and denominator are polynomials
• Example: Given $f(x) = \frac{3x-1}{2x+1}$ and $g(x) = \frac{2x+3}{x-2}$, find $(f+g)(x)$
  1. $(f+g)(x) = \frac{3x-1}{2x+1} + \frac{2x+3}{x-2}$
  2. Find the LCD: $(2x+1)(x-2)$
  3. Rewrite the expressions with the LCD: $\frac{(3x-1)(x-2)}{(2x+1)(x-2)} + \frac{(2x+3)(2x+1)}{(x-2)(2x+1)}$
  4. Simplify the numerators: $\frac{3x^2-7x+2}{(2x+1)(x-2)} + \frac{4x^2+7x+3}{(x-2)(2x+1)}$
  5. Add the numerators and keep the LCD: $\frac{3x^2-7x+2+4x^2+7x+3}{(2x+1)(x-2)} = \frac{7x^2+5}{(2x+1)(x-2)}$
• The simplified result is $(f+g)(x) = \frac{7x^2+5}{(2x+1)(x-2)}$

Domain restrictions

• When working with rational expressions, it's crucial to consider domain restrictions
• The denominator of a rational expression cannot equal zero, as division by zero is undefined
• Identify values that make the denominator zero and exclude them from the domain of the rational expression
• These restrictions must be considered when simplifying or combining rational expressions to ensure the final result is valid for all permissible input values