Final answer:
To factor the equation f(x) = x³ - 6x² - 4x - 24, we can start by checking if any small integers are roots of the equation. By trying x = -1, we find that (x + 1) is a factor. Dividing the equation by (x + 1), we get the remaining quadratic equation x² - 7x - 24. Factoring this quadratic equation, we get (x - 8)(x + 3). Therefore, the factored form is f(x) = (x + 1)(x - 8)(x + 3).
Step-by-step explanation:
The given equation is f(x) = x³ - 6x² - 4x - 24. To factor this equation, we need to find the roots of the equation, which are the values of x that make the equation equal to zero.
We can start by checking if any small integers are roots of the equation. By trying x = -1, we find that f(-1) = 0. Therefore, (x + 1) is a factor of the equation.
Using synthetic division or long division, we can divide the equation f(x) by (x + 1) to get the remaining quadratic equation x² - 7x - 24. Factoring this quadratic equation, we get (x - 8)(x + 3).
So, the factored form of the equation f(x) = x³ - 6x² - 4x - 24 is f(x) = (x + 1)(x - 8)(x + 3).