%2F(x%5E2%2B2x%2B1).jpeg "(x^3-x^2-5x-3)/(x^2+2x+1)")
Simplifying the Rational Expression: (x^3 - x^2 - 5x - 3) / (x^2 + 2x + 1)
This article will walk through the process of simplifying the rational expression (x^3 - x^2 - 5x - 3) / (x^2 + 2x + 1).
Factoring the Expressions
The first step is to factor both the numerator and the denominator.
- Numerator: (x^3 - x^2 - 5x - 3)
We can use the Rational Root Theorem to find potential roots of this polynomial. The theorem states that any rational root must be of the form p/q, where p is a factor of the constant term (-3) and q is a factor of the leading coefficient (1). Therefore, the possible rational roots are ±1, ±3.
By testing these values, we find that x = -1 is a root of the polynomial. This means (x + 1) is a factor. Performing polynomial division, we find that:
(x^3 - x^2 - 5x - 3) = (x + 1)(x^2 - 2x - 3)
We can further factor the quadratic expression:
(x^2 - 2x - 3) = (x - 3)(x + 1)
Therefore, the factored numerator is: (x + 1)(x - 3)(x + 1)
- Denominator: (x^2 + 2x + 1)
This is a perfect square trinomial:
(x^2 + 2x + 1) = (x + 1)^2
Simplifying the Expression
Now we can rewrite the original expression:
(x^3 - x^2 - 5x - 3) / (x^2 + 2x + 1) = [(x + 1)(x - 3)(x + 1)] / [(x + 1)^2]
Notice that we have a common factor of (x + 1) in both the numerator and denominator. We can cancel this out, leaving us with:
(x - 3)(x + 1) / (x + 1)
Finally, we can cancel another (x + 1) factor:
x - 3
Conclusion
By factoring and simplifying, we determined that the expression (x^3 - x^2 - 5x - 3) / (x^2 + 2x + 1) simplifies to x - 3. Remember that this simplification is only valid when x ≠ -1, as the original expression is undefined at x = -1.