X3-5-14x/(x²-2x-8)

To simplify the expression \(\frac{x^3 - 5 - 14x}{x^2 - 2x - 8}\), we can start by factoring both the numerator and the denominator if possible. 1. **Factor the denominator \(x^2 - 2x - 8\)**: We need to find two numbers that multiply to \(-8\) (the constant term) and add to \(-2\) (the coefficient of the linear term). The two numbers that satisfy this are \( -4 \) and \( 2 \). Thus, we can factor the denominator as: \[ x^2 - 2x - 8 = (x - 4)(x + 2) \] 2. **Simplify the numerator \(x^3 - 5 - 14x\)**: We can rearrange the numerator as: \[ x^3 - 14x - 5 \] To factor this, we can use synthetic division or the Rational Root Theorem to check for possible roots. Trying \(x = 3\): \[ 3^3 - 14(3) - 5 = 27 - 42 - 5 = -20 \quad \text{(not a root)} \] Trying \(x = -1\): \[ (-1)^3 - 14(-1) - 5 = -1 + 14 - 5 = 8 \quad \text{(not a root)} \] Trying \(x = -5\): \[ (-5)^3 - 14(-5) - 5 = -125 + 70 - 5 = -60 \quad \text{(not a root)} \] Since finding rational roots isn't yielding results easily, we could also consider performing polynomial long division, but since this might be more complex, we'll choose the numerical approach: graphical or numerical methods or an approximate polynomial alteration. Without further simplifications from roots, we may analyze the division directly. For basic polynomial long division, performing it would yield certain terms as follows, but we will focus on the original factorization. Because there is no evident factoring of the numerator that further samples simplification that matches the denominator, we can focus on conventional division of fractions. ### Result So in conclusion, the simplified term remains: \[ \frac{x^3 - 14x - 5}{(x - 4)(x + 2)} \] This indicates no factors among the numerator divide \( (x - 4)(x + 2) \) thoroughly. The expression is now properly simplified.