Mathematics

The zeros of this function f(x) are the values of x for which the function produces a result of zero. We can use the quadratic formula to solve this equation for the zeros. First we factor the leading coefficient (1):

f(x) = x^3 - 8x^2 + 9x + 18
= (x - 1)(x^2 - 7x + 18)

Next, we can use the quadratic formula to find the remaining factors (x = 3, 6):

x = [-b +/- sqrt(b^2 - 4ac)] / 2a

Plugging in the values for a, b, and c, we get the following:

x = [-(-7) +/- sqrt((-7)^2 - 4(1)(18)] / 2(1)

x = [7 +/- sqrt(49 - 72)] / 2

x = [7 +/- sqrt(-23)] / 2

Since there is no real number solution for the square root of a negative number, the only real zeros for this function are x = -1, 3, and 6.

The complete factored form of f(x) is f(x) = (x + 1)(x - 3)(x - 6).

The factors of f(x) reflect the three zeros of the function: -1, 3 and 6. This is because a factor (x + a) will produce a zero at x = -a and a factor (x - a) will produce a zero at x = +a. Since the three given zeros are -1, 3 and 6, each of these is included in the factored form in the form of a factor.