Determine an equation, in simplified form, for the family of cubic functions with zeros 2, -sqrt3, +sqrt3 and -4

I solved it

Just for your information:

you called for a cubic, but you have 4 zeros, so your function would be a
quartic:
f(x) = (x-1)(x^2 - 3)(x+4)

To determine the equation for the family of cubic functions with the given zeros, we need to use the fact that zeros of a cubic function are also the roots of its corresponding polynomial equation.

From the given zeros:
Zero 1: 2
Zero 2: -√3
Zero 3: √3
Zero 4: -4

We can represent each zero as a factor of the cubic equation:
(x - 2)(x + √3)(x - √3)(x + 4) = 0

Expanding the equation:
(x - 2)(x^2 - 3)(x + 4) = 0

To obtain the simplified form, we can multiply the factors:
(x^2 - 2x√3 - 3)(x + 4) = 0

Now, we can multiply the binomial expressions:
x^3 + 4x^2 - 2x^2√3 - 8x - 3x√3 - 12 = 0

Combining like terms, we get the simplified equation for the family of cubic functions with zeros 2, -√3, +√3, and -4:

x^3 + 2x^2 - 11x - 12 + (-3√3 - 2√3)x = 0

or in simplified form:

x^3 + 2x^2 - 11x - 12 - 5√3x = 0

To determine a cubic equation with given zeros, we can use the zero-product property. Since the zeros are 2, -√3, +√3, and -4, we can write the equation as follows:

(x - 2)(x + √3)(x - √3)(x + 4) = 0

Let's simplify this equation step by step:

1. Multiply (x - 2) and (x + 4):

(x^2 + 4x - 2x - 8) * (x + √3)(x - √3) = 0

(x^2 + 2x - 8) * (x + √3)(x - √3) = 0

2. Expand (x + √3)(x - √3):

(x^2 + 2x - 8) * (x^2 - (√3)^2) = 0

(x^2 + 2x - 8) * (x^2 - 3) = 0

3. Distribute the remaining factors:

x^4 - 3x^2 + 2x^3 - 6x - 8x^2 + 24 = 0

4. Combine like terms:

x^4 + 2x^3 - 11x^2 - 6x + 24 = 0

Therefore, the equation for the family of cubic functions with zeros 2, -√3, +√3, and -4 is:

f(x) = x^4 + 2x^3 - 11x^2 - 6x + 24