Identify the simplest polynomial function having integer coefficients with the given zeros 0, -4, square root 3

Question

a. P(x) = x⁴ + 4x³ − 3x² − 12x

b. P(x) = x³ + 4x² − 12x − 3

c. P(x) = x³ + 4x² − 3x − 12

d. P(x) = x⁴ + 4x³ − 12x² − 3x

Answer 1

Remember that when you are trying to find the zeros of a function, you can factor it and then set each factor equal to zero to find them. So, if you have the zeros, you can work backwards to get the factors. So, if x = 0, x = -4, and x = √3 are zeros, then (x - 0), (x + 4), and (x - √3) are factors. However, if we multiply these factors together to get a polynomial, it will not have integer coefficients because of the √3. So, to fix that, we can add another "opposite" factor, also known as a conjugate, to (x - √3) which would be (x + √3).

So P(x) = (x - 0)(x + 4)(x - √3)(x + √3)

You can multiply (x - √3)(x + √3) together using FOIL to get x² - 3.

You can multiply (x - 0)(x + 4) together to get x² + 4x.

Now multiply (x² - 3) and (x² + 4x) to get the final result.