Final answer:
To find the zeros of the polynomial function f(x) = x³(x-3)²(x+6), you simply set each factor equal to zero. This gives zeros at x = 0 with multiplicity 3, x = 3 with multiplicity 2, and x = -6 with multiplicity 1.
Step-by-step explanation:
To find the zeros of the polynomial function f(x) = x³(x-3)²(x+6), you need to set the function equal to zero and solve for x. The zeros of a polynomial are the values of x for which the polynomial equals zero. These occur when any of the factors in the polynomial are zero.
Since the factors are x, x-3, and x+6, the zeros can be found by setting each factor to zero:
- x = 0
- x - 3 = 0, which gives x = 3
- x + 6 = 0, which gives x = -6
This polynomial has a zero at x = 0 with a multiplicity of 3, since the x term is cubed, a zero at x = 3 with a multiplicity of 2, since the (x-3) term is squared, and a single zero at x = -6.