(x^3+3x^2-4x-12)/(x^2+5x+6)

2 min read Jun 17, 2024
(x^3+3x^2-4x-12)/(x^2+5x+6)

Simplifying Rational Expressions: (x^3+3x^2-4x-12)/(x^2+5x+6)

This article will explore the process of simplifying the rational expression (x^3+3x^2-4x-12)/(x^2+5x+6).

1. Factor the numerator and denominator.

  • Numerator:
    • We can factor by grouping:
    • (x^3 + 3x^2) + (-4x - 12)
    • x^2(x + 3) - 4(x + 3)
    • (x + 3)(x^2 - 4)
    • We can further factor (x^2 - 4) as a difference of squares:
    • (x + 3)(x + 2)(x - 2)
  • Denominator:
    • We can factor the quadratic:
    • (x + 2)(x + 3)

2. Identify common factors

Now our expression looks like this: [(x + 3)(x + 2)(x - 2)] / [(x + 2)(x + 3)] We can see that both the numerator and denominator share the factors (x + 3) and (x + 2).

3. Simplify by canceling common factors.

We can cancel out the common factors, leaving us with: (x - 2) / 1

4. Final Simplified Expression

The simplified form of the rational expression (x^3+3x^2-4x-12)/(x^2+5x+6) is (x - 2).

Important Note: It's crucial to remember that the original expression and the simplified one are equivalent except when x = -3 or x = -2. These values make the original denominator zero, rendering the expression undefined. The simplified form doesn't reflect this restriction.

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