Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions

Objectives

• Adding and subtracting rational expressions with like denominators.
• Adding and subtracting rational expressions with unlike denominators.
• Always write the fraction in the simplest form.

Adding and Subtracting with Like Denominators

• If the denominators of fractions are the same, add or subtract the numerators.

Example with Rational Numbers

• Adding fractions
  • $\frac{1}{5} + \frac{7}{5} = \frac{1+7}{5} = \frac{8}{5}$
  • $\frac{1}{5}$ is a proper fraction (numerator < denominator).
  • $\frac{7}{5}$ is an improper fraction (numerator > denominator).
• Subtracting fractions
  • $\frac{1}{5} - \frac{7}{5} = \frac{1-7}{5} = \frac{-6}{5}$
  • Integers: negative counting numbers, zero, and positive counting numbers.

Example with Algebraic Expressions

• Example 1:
  • $\frac{x}{4} + \frac{5-x}{4}$
  • Denominators are the same.
  • Add the numerators: $\frac{x + 5 - x}{4}$
  • Simplify: $\frac{5}{4}$
  • Domain: All real numbers because the denominator (4) is different than zero, and there is no x in the denominator.
  • Interval notation: $(-\infty, \infty)$
• Example 2:
  • $\frac{x}{x^2 - 2x - 3} - \frac{3}{x^2 - 2x - 3}$
  • Factor the denominator: $x^2 - 2x - 3 = (x - 3)(x + 1)$
    • Possible combinations for -3: -3 and 1, since $-3 + 1 = -2$
  • Rewrite the expression:
    • $\frac{x}{(x - 3)(x + 1)} - \frac{3}{(x - 3)(x + 1)}$
  • Subtract the numerators: $\frac{x - 3}{(x - 3)(x + 1)}$
  • Simplify: $\frac{1}{x + 1}$
  • Domain: Use the original factored form of the denominator, $(x - 3)(x + 1) \neq 0$
    • $x \neq 3$ and $x \neq -1$
    • Interval notation: $(-\infty, -1) \cup (-1, 3) \cup (3, \infty)$

Least Common Multiple (LCM)

Finding the LCM of Algebraic Expressions

• Monomials: Find the LCM of the coefficients and the highest power of the variable.
  • Example: Find the LCM of $6x, 2x^2, 9x^3$
    • Variable: $x^3$
    • Coefficients: 6, 2, and 9.
      • Multiples of 6: 6, 12, 18, 24, …
      • Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, …
      • Multiples of 9: 9, 18, 27, …
    • LCM of coefficients: 18
    • LCM of the algebraic expressions: $18x^3$
• Binomials: Factorize the expressions first.
  • Example: Find the LCM of $x^2 - x$ and $2x - 2$
    • Factorization:
      • $x^2 - x = x(x - 1)$
      • $2x - 2 = 2(x - 1)$
    • Factors: 2, x, and x - 1.
    • LCM: $2 * x * (x-1) = 2x(x - 1)$
• Trinomials: Factorize the expressions first.
  • Example: Find the LCM of $3x^2 + 6x$ and $x^2 + 4x + 4$
    • Factorization:
      • $3x^2 + 6x = 3x(x + 2)$
      • $x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2$
    • Factors: 3x and x + 2.
    • LCM: $3x(x + 2)^2$

Adding and Subtracting with Unlike Denominators

• Find the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the LCM as the denominator.
• Add or subtract the numerators.
• Simplify the result.

Example 1

• $\frac{7}{6x} + \frac{5}{8x}$
• LCM of 6x and 8x:
  • LCM of 6 and 8: 24
  • LCM of x and x: x
  • LCM: 24x
• Convert fractions:
  • $\frac{7}{6x} * \frac{4}{4} = \frac{28}{24x}$
  • $\frac{5}{8x} * \frac{3}{3} = \frac{15}{24x}$
• Add the fractions: $\frac{28}{24x} + \frac{15}{24x} = \frac{28 + 15}{24x} = \frac{43}{24x}$

Example 2

• $\frac{2x}{x + 1} - \frac{x}{x + 2}$
  • x cannot be -1 and -2 (given).
• LCM of x + 1 and x + 2: (x + 1)(x + 2)
• Convert fractions:
  • $\frac{2x}{x + 1} * \frac{x + 2}{x + 2} = \frac{2x(x + 2)}{(x + 1)(x + 2)} = \frac{2x^2 + 4x}{(x + 1)(x + 2)}$
  • $\frac{x}{x + 2} * \frac{x + 1}{x + 1} = \frac{x(x + 1)}{(x + 1)(x + 2)} = \frac{x^2 + x}{(x + 1)(x + 2)}$
• Subtract the fractions:
  • $\frac{2x^2 + 4x}{(x + 1)(x + 2)} - \frac{x^2 + x}{(x + 1)(x + 2)} = \frac{2x^2 + 4x - (x^2 + x)}{(x + 1)(x + 2)} = \frac{2x^2 + 4x - x^2 - x}{(x + 1)(x + 2)} = \frac{x^2 + 3x}{(x + 1)(x + 2)}$
  • Domain: x cannot be -1 or -2.

Example 3

• $\frac{3}{x^2 - 1} + \frac{x}{x^2 + 2x + 1}$
• Factorize denominators:
  • $x^2 - 1 = (x + 1)(x - 1)$
  • $x^2 + 2x + 1 = (x + 1)^2$
• LCM: $(x + 1)^2(x - 1)$
• Convert fractions:
  • $\frac{3}{(x + 1)(x - 1)} * \frac{x + 1}{x + 1} = \frac{3(x + 1)}{(x + 1)^2(x - 1)}$
  • $\frac{x}{(x + 1)^2} * \frac{x - 1}{x - 1} = \frac{x(x - 1)}{(x + 1)^2(x - 1)}$
• Add the fractions:
  • $\frac{3(x + 1)}{(x + 1)^2(x - 1)} + \frac{x(x - 1)}{(x + 1)^2(x - 1)} = \frac{3x + 3 + x^2 - x}{(x + 1)^2(x - 1)} = \frac{x^2 + 2x + 3}{(x + 1)^2(x - 1)}$
  • Domain: x cannot be -1 or 1.

Example 4

• $\frac{1}{x^2 - x} - \frac{3x}{4x^2 - 4x}$
• Simplify $4x^2 - 4x = 4x(x-1)$
• LCM between $x^2 - x = x(x-1)$ and $4x(x-1)$ is $4x(x-1)^2$
  • x must be different than 0 and 1.
• Simplify the expression to find the original form: $\frac{x^2 + 4x - 8}{4x(x - 1)^2}$

Complex Fractions

• A fraction over another fraction.
• Represents the division of fractions.
• Example: $\frac{\frac{1}{2} + \frac{1}{x}}{\frac{x^2 - 4}{2x}}$

Simplifying Complex Fractions

• Simplify the numerator and denominator separately.
• Rewrite the complex fraction as a division problem.
• Multiply by the reciprocal of the denominator.
• Simplify the result.

Example

• $\frac{\frac{1}{2} + \frac{1}{x}}{\frac{x^2 - 4}{2x}}$
• Simplify the numerator:
  • $\frac{1}{2} + \frac{1}{x} = \frac{x}{2x} + \frac{2}{2x} = \frac{x + 2}{2x}$
• Rewrite the complex fraction as a division problem:
  • $\frac{x + 2}{2x} \div \frac{x^2 - 4}{2x}$
• Multiply by the reciprocal of the denominator:
  • $\frac{x + 2}{2x} * \frac{2x}{x^2 - 4} = \frac{x + 2}{2x} * \frac{2x}{(x + 2)(x - 2)}$
• Simplify the result:
  • $\frac{x + 2}{2x} * \frac{2x}{(x + 2)(x - 2)} = \frac{1}{x - 2}$
  • Restrictions: x cannot be 0, -2, or 2.