To find the zeros of the polynomial function f(x) = x³ – 9x² – 20x, we need to set the function equal to zero:
x³ – 9x² – 20x = 0
Next, we can factor out the common term, which in this case is x:
x(x² – 9x – 20) = 0
This gives us one zero immediately: x = 0. After factoring out x, we now need to solve the quadratic equation x² – 9x – 20 = 0.
We can factor the quadratic expression:
x² – 9x – 20 = (x – 10)(x + 2) = 0
Setting each factor equal to zero gives us:
- x – 10 = 0 → x = 10
- x + 2 = 0 → x = -2
In summary, the zeros of the polynomial function f(x) = x³ – 9x² – 20x are:
- x = 0
- x = 10
- x = -2
These values are where the function crosses the x-axis, and they are essential for understanding the behavior of the polynomial.