Difference of Rational Expressions: Subtraction and Domain

Overview

• Problem: Subtract two rational expressions:
  • \frac{a-2}{a+2} - \frac{a-3}{a^2+4a+4}
• Goal: express the result as a simplified rational expression and state the domain.

Key ideas

• To subtract fractions, need a common denominator (LCM of denominators).
• Factor denominators to find common denominator.

Factorization

• The first denominator: a+2 (already linear).
• The second denominator: a^2+4a+4
• Recognize pattern: a^2+4a+4 = (a+2)^2 (since 4 is 2^2 and cross term 4a).
• Therefore LCM (least common multiple) of denominators is (a+2)^2
• So the common denominator is (a+2)^2

Setup for subtraction

• Rewrite first fraction with common denominator:

  • Multiply numerator and denominator by a+2:
  • \frac{a-2}{a+2} \cdot \frac{a+2}{a+2} = \frac{(a-2)(a+2)}{(a+2)^2} = \frac{a^2-4}{(a+2)^2}

• The second fraction already has common denominator:

  • \frac{a-3}{a^2+4a+4} = \frac{a-3}{(a+2)^2}

• Domain note: Need to exclude values that make any denominator zero, i.e., a \neq -2. So domain is all real numbers except -2.

Combine and simplify

• Subtract numerators over the common denominator:
  • \frac{a^2-4}{(a+2)^2} - \frac{a-3}{(a+2)^2} = \frac{(a^2-4) - (a-3)}{(a+2)^2}
• Simplify numerator:
  • (a^2-4) - (a-3) = a^2 - 4 - a + 3 = a^2 - a - 1
• Result:
  • \frac{a^2 - a - 1}{(a+2)^2}

Further simplification check

• Check if numerator shares a factor with the denominator:
  • Denominator factors: (a+2)^2
  • Check if a+2 divides numerator a^2 - a - 1 .
  • Evaluate at a = -2 : (-2)^2 - (-2) - 1 = 4 + 2 - 1 = 5 \neq 0 so not a factor.
  • Hence no common factor; the fraction is in lowest terms.

Domain considerations

• Denominator equals zero when a = -2 , leading to undefined expression.
• Therefore the domain is all real numbers with a \neq -2 .
• Note: The second denominator originally is a^2+4a+4 , which is also zero at a = -2, consistent.

Final result

• Simplified expression:
  • \frac{a^2 - a - 1}{(a+2)^2}
• Domain:
  • a \in \mathbb{R}, \quad a \neq -2
• Quick check (optional): pick a value e.g., a = 0:
  • Original: \frac{-2}{2} - \frac{-3}{4} = -1 + \frac{3}{4} = -\frac{1}{4}
  • Final: \frac{0 - 0 - 1}{(0+2)^2} = \frac{-1}{4} OK.

Connections and comments

• Demonstrates standard technique for subtracting rational expressions: use common denominator via factoring.
• Emphasizes domain restrictions from denominators.
• Similar approach to addition of rational expressions.