If f is a polynomial function of degree 3 with the numbers 4,5, and -6 as zeros, and if f(1)=-210, determine the formula for f(x).

let the equation be

y = a(x-4)(x-5)(x+6)
f(1) = -210
-210 = a(-3)(-4)(7)
a = -210/84 = -5/2

f(x) = (-5/2)(x-4)(x-5)(x+6)

To find the formula for the polynomial function f(x), we need to know the zeros of the function. The zeros of f(x) are the values of x for which f(x) equals zero.

In this case, we are given that the zeros of f(x) are 4, 5, and -6. This means that when x = 4, 5, or -6, f(x) = 0.

We can write the equation for f(x) in factored form using the zeros:

f(x) = a(x - 4)(x - 5)(x + 6),

where a is a constant coefficient.

Next, we are given that f(1) = -210. Plugging in x = 1 into the equation, we have:

-210 = a(1 - 4)(1 - 5)(1 + 6).

Simplifying the equation, we get:

-210 = a(-3)(-4)(7).

-210 = 84a.

Dividing both sides of the equation by 84, we find:

a = -210/84.

Simplifying this further, we get:

a = -5/2.

Now that we have the value of a, we can substitute it back into the equation for f(x):

f(x) = (-5/2)(x - 4)(x - 5)(x + 6).

Therefore, the formula for f(x) is:

f(x) = (-5/2)(x - 4)(x - 5)(x + 6).