This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — It appears there might have been a slight typo in the original problem. The polynomial x^3 - 2x^2 - 5x - 6 does not have any rational roots, meaning it cannot be factored into linear factors with rational coefficients. Assuming the problem intended to be x^3 - 2x^2 - 5x + 6 (a common variation that is factorable with integer roots), here's how to factor it: Step 1: Find a root of the polynomial using the Rational Root Theorem. Let P(x) = x^3 - 2x^2 - 5x + 6. The possible rational roots are factors of the constant term (6) divided by factors of the leading coefficient (1). Factors of 6: ± 1, ± 2, ± 3, ± 6. Let's test x=1: P(1) = (1)^3 - 2(1)^2 - 5(1) + 6 = 1 - 2 - 5 + 6 = -1 - 5 + 6 = -6 + 6 = 0 Since P(1) = 0, x=1 is a root, which means (x-1) is a factor of P(x). Step 2: Use synthetic division to divide P(x) by (x-1). The coefficients of P(x) are 1, -2, -5, 6. c|cccc 1 & 1 & -2 & -5 & 6 \\ & & 1 & -1 & -6 \\ & 1 & -1 & -6 & 0 \\ The quotient is x^2 - x - 6. Step 3: Factor the resulting quadratic expression. Now we need to factor x^2 - x - 6. We look for two numbers that multiply to -6 and add to -1. These numbers are -3 and 2. So, x^2 - x - 6 = (x-3)(x+2). Step 4: Write the fully factored polynomial. Combine the factor from Step 1 and the factors from Step 3: x^3 - 2x^2 - 5x + 6 = (x-1)(x-3)(x+2) The factored form (assuming the typo correction) is (x-1)(x-3)(x+2). What's next? 📸