To find the zeros of the polynomial function f(x) = x³ – 2x² – 24x, we need to set the function equal to zero and solve for x:
f(x) = 0
=> x³ – 2x² – 24x = 0
First, we can factor out the common term, which is x:
x(x² – 2x – 24) = 0
This gives us one zero right away: x = 0. Now, we need to solve the quadratic equation x² – 2x – 24 = 0.
To factor the quadratic, we look for two numbers that multiply to -24 and add to -2. These numbers are -6 and 4. So we can factor the quadratic as:
(x – 6)(x + 4) = 0
Now, we set each factor equal to zero:
x – 6 = 0
=> x = 6
x + 4 = 0
=> x = -4
To summarize, the zeros of the polynomial function are:
- x = 0
- x = 6
- x = -4