To find the zeros of the polynomial function f(x) = x³ – 9x² – 20x, we need to determine the values of x for which f(x) = 0.
Step 1: Factor the polynomial. We can start by factoring out the common term from the expression:
f(x) = x(x² – 9x – 20)
Step 2: Now, we need to factor the quadratic x² – 9x – 20. We will look for two numbers that multiply to give -20 and add up to give -9.
The numbers that satisfy this condition are -10 and 2. Thus, we can factor the quadratic:
x² – 9x – 20 = (x – 10)(x + 2)
Step 3: Now substituting this back into the factored form of f(x) gives us:
f(x) = x(x – 10)(x + 2)
Step 4: Setting f(x) equal to zero and solving for x gives us:
x(x – 10)(x + 2) = 0
This equation is satisfied when any of its factors equals zero:
- x = 0
- x – 10 = 0 → x = 10
- x + 2 = 0 → x = -2
Step 5: Thus, the zeros of the polynomial function f(x) = x³ – 9x² – 20x are:
- x = 0
- x = 10
- x = -2
In summary, the zeros of the given polynomial function are 0, 10, and -2.