Well, isn't this equation going to be a real "rootin' tootin'" time! Let's start by recalling that if a cubic function has zeros at x = a, b, and c, then its factored form can be written as (x-a)(x-b)(x-c).
In this case, the zeros are 2, 4 + sqrt(3), and 4 - sqrt(3). To simplify things a bit, let's call the latter two zeros A and B respectively.
So, our factored form would be (x - 2)(x - A)(x - B). Now, to find A and B, we need to determine their values based on the given information.
If 4 + sqrt(3) is a zero, then (x - (4 + sqrt(3))) = 0. Simplifying that, we get x - 4 - sqrt(3) = 0. Rearranging, we find that x - (4 - sqrt(3)) = 0. This means that 4 - sqrt(3) is our second zero, B.
Now we have our factored form: (x - 2)(x - (4 + sqrt(3)))(x - (4 - sqrt(3))).
But, we're not finished yet! Let's expand this equation and simplify it further.
(x - 2)(x - (4 + sqrt(3)))(x - (4 - sqrt(3)))
(x - 2)(x - 4 - sqrt(3))(x - 4 + sqrt(3))
(x - 2)((x - 4) - sqrt(3))((x - 4) + sqrt(3))
(x - 2)((x - 4)^2 - (sqrt(3))^2)
(x - 2)(x^2 - 8x + 16 - 3)
(x - 2)(x^2 - 8x + 13)
And there you have itβan equation for the family of cubic functions with zeros 2, 4 + sqrt(3), and 4 - sqrt(3): f(x) = (x - 2)(x^2 - 8x + 13).