To find the zeros of the polynomial function f(x) = x³ + x² – 20x, we need to set the function equal to zero:
f(x) = x³ + x² – 20x = 0.
First, we can factor out the greatest common factor, which in this case is x:
x(x² + x – 20) = 0.
This gives us one zero immediately, which is x = 0.
Next, we need to solve the quadratic equation inside the parentheses:
x² + x – 20 = 0.
We can factor this quadratic as well. We need two numbers that multiply to -20 and add up to 1. These numbers are 5 and -4. So, we can write:
(x + 5)(x – 4) = 0.
Now, setting each factor equal to zero gives us the other zeros:
x + 5 = 0 → x = -5
x – 4 = 0 → x = 4
In summary, the zeros of the polynomial function f(x) = x³ + x² – 20x are:
- x = 0
- x = -5
- x = 4
These are the values of x where the polynomial intersects the x-axis.