Adding and Subtracting Rational Expressions

Overview

Adding and subtracting rational expressions involves working with fractions where both the numerator and denominator are polynomials. To perform these operations, we need a common denominator, just like with simple fractions. The process can sometimes involve factoring or finding the least common denominator (LCD) to combine the terms.

Adding Rational Expressions

To add rational expressions, follow these steps:

  1. Factor both the numerators and denominators of the expressions.
  2. Find the least common denominator (LCD) of the two expressions.
  3. Rewrite both expressions with the LCD as the denominator.
  4. Add the numerators together.
  5. Simplify the expression, if possible.

Example 1: Adding Rational Expressions

Consider the following addition problem:

\( \frac{x}{x^2 - 4} + \frac{2}{x^2 - 4} \)

Step 1: Factor the denominators:

\( \frac{x}{(x - 2)(x + 2)} + \frac{2}{(x - 2)(x + 2)} \)

Step 2: The denominators are already the same, so we can proceed to add the numerators:

\( \frac{x + 2}{(x - 2)(x + 2)} \)

Step 3: Simplify the expression if possible (in this case, it cannot be simplified further). The result is:

\( \frac{x + 2}{(x - 2)(x + 2)} \)

Subtracting Rational Expressions

To subtract rational expressions, follow the same steps as adding, but subtract the numerators instead:

  1. Factor both the numerators and denominators of the expressions.
  2. Find the least common denominator (LCD) of the two expressions.
  3. Rewrite both expressions with the LCD as the denominator.
  4. Subtract the numerators.
  5. Simplify the expression, if possible.

Example 2: Subtracting Rational Expressions

Consider the following subtraction problem:

\( \frac{3}{x^2 - 4} - \frac{1}{x^2 - 4} \)

Step 1: Factor the denominators:

\( \frac{3}{(x - 2)(x + 2)} - \frac{1}{(x - 2)(x + 2)} \)

Step 2: The denominators are already the same, so we can proceed to subtract the numerators:

\( \frac{3 - 1}{(x - 2)(x + 2)} \)

Step 3: Simplify the numerator:

\( \frac{2}{(x - 2)(x + 2)} \)

Practice Questions

  1. Question 1: Add the following rational expressions:

    \( \frac{5}{x^2 - 9} + \frac{3}{x^2 - 9} \)

    Solution

    Step 1: Factor the denominators:

    \( \frac{5}{(x - 3)(x + 3)} + \frac{3}{(x - 3)(x + 3)} \)

    Step 2: The denominators are already the same, so we can add the numerators:

    \( \frac{5 + 3}{(x - 3)(x + 3)} \)

    Step 3: Simplify the numerator:

    \( \frac{8}{(x - 3)(x + 3)} \)

  2. Question 2: Subtract the following rational expressions:

    \( \frac{7}{x^2 - 4} - \frac{3}{x^2 - 4} \)

    Solution

    Step 1: Factor the denominators:

    \( \frac{7}{(x - 2)(x + 2)} - \frac{3}{(x - 2)(x + 2)} \)

    Step 2: The denominators are already the same, so we can subtract the numerators:

    \( \frac{7 - 3}{(x - 2)(x + 2)} \)

    Step 3: Simplify the numerator:

    \( \frac{4}{(x - 2)(x + 2)} \)

  3. Question 3: Add the following rational expressions:

    \( \frac{x}{x^2 - 4x + 3} + \frac{2}{x^2 - 4x + 3} \)

    Solution

    Step 1: Factor the denominators:

    \( \frac{x}{(x - 1)(x - 3)} + \frac{2}{(x - 1)(x - 3)} \)

    Step 2: The denominators are already the same, so we can add the numerators:

    \( \frac{x + 2}{(x - 1)(x - 3)} \)

  4. Question 4: Subtract the following rational expressions:

    \( \frac{5}{x^2 - x - 6} - \frac{1}{x^2 - x - 6} \)

    Solution

    Step 1: Factor the denominators:

    \( \frac{5}{(x - 3)(x + 2)} - \frac{1}{(x - 3)(x + 2)} \)

    Step 2: The denominators are already the same, so we can subtract the numerators:

    \( \frac{5 - 1}{(x - 3)(x + 2)} \)

    Step 3: Simplify the numerator:

    \( \frac{4}{(x - 3)(x + 2)} \)

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