Final answer:
The polynomial x^3 - 2x^2 - 24x can be factored by first taking out the common factor x, and then factoring the resulting quadratic to get x(x - 6)(x + 4). The correct answer is option b).
Step-by-step explanation:
Factoring the Polynomial
To factor the polynomial x^3 - 2x^2 - 24x, we first look for a common factor in all terms. We can see that x is a common factor. Factoring out x, we get x(x^2 - 2x - 24). Now, we need to factor the quadratic equation inside the parentheses, which has the form ax^2 + bx + c.
To find factors of the quadratic, we look for two numbers that multiply to give ac (the product of the coefficient of the x^2 term and the constant term, which is -24) and add to give b (the coefficient of the x term, which is -2). The numbers that fit this requirement are -6 and +4. This gives us the factors (x - 6) and (x + 4). Therefore, the fully factored form of the polynomial is x(x - 6)(x + 4).
The correct answer from the options given is b) x(x + 4)(x - 6).