%2F(x%5E2%2B5x%2B6).jpeg "(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.