Difference of Rational Expressions: Subtraction and Domain

Overview

  • Problem: Subtract two rational expressions:
    • a−2a+2−a−3a2+4a+4\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+2a+2 (already linear).
  • The second denominator: a2+4a+4a^2+4a+4
  • Recognize pattern: a2+4a+4=(a+2)2a^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(a+2)^2
  • So the common denominator is (a+2)2(a+2)^2

Setup for subtraction

  • Rewrite first fraction with common denominator:

    • Multiply numerator and denominator by a+2a+2:
    • a−2a+2⋅a+2a+2=(a−2)(a+2)(a+2)2=a2−4(a+2)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:

    • a−3a2+4a+4=a−3(a+2)2\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:
    • a2−4(a+2)2−a−3(a+2)2=(a2−4)−(a−3)(a+2)2\frac{a^2-4}{(a+2)^2} - \frac{a-3}{(a+2)^2} = \frac{(a^2-4) - (a-3)}{(a+2)^2}
  • Simplify numerator:
    • (a2−4)−(a−3)=a2−4−a+3=a2−a−1(a^2-4) - (a-3) = a^2 - 4 - a + 3 = a^2 - a - 1
  • Result:
    • a2−a−1(a+2)2\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(a+2)^2
    • Check if a+2a+2 divides numerator a2−a−1a^2 - a - 1.
    • Evaluate at a=−2a = -2: (−2)2−(−2)−1=4+2−1=5≠0(-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=−2a = -2, leading to undefined expression.
  • Therefore the domain is all real numbers with a≠−2a \neq -2.
  • Note: The second denominator originally is a2+4a+4a^2+4a+4, which is also zero at a = -2, consistent.

Final result

  • Simplified expression:
    • a2−a−1(a+2)2\frac{a^2 - a - 1}{(a+2)^2}
  • Domain:
    • a∈R,a≠−2a \in \mathbb{R}, \quad a \neq -2
  • Quick check (optional): pick a value e.g., a = 0:
    • Original: −22−−34=−1+34=−14\frac{-2}{2} - \frac{-3}{4} = -1 + \frac{3}{4} = -\frac{1}{4}
    • Final: 0−0−1(0+2)2=−14\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.