To find the zeros of the polynomial function f(x) = x³ + x² – 12x, we first set the function equal to zero:
x³ + x² – 12x = 0
Next, we can factor the equation. Notice that there is a common factor of x in each term:
x(x² + x – 12) = 0
Now, we can set each factor equal to zero. The first factor gives us:
x = 0
For the second factor, we need to solve the quadratic equation x² + x – 12 = 0. We can factor this further:
(x + 4)(x – 3) = 0
Setting each factor equal to zero gives us:
x + 4 = 0 ⇒ x = -4
x – 3 = 0 ⇒ x = 3
Thus, the zeros of the polynomial function f(x) are:
- x = 0
- x = -4
- x = 3
These values are where the function intersects the x-axis, meaning that f(x) = 0 at these points. Therefore, the complete set of zeros for the polynomial function is {0, -4, 3}.