Question

What is the simplest polynomial function `f(x)` with zeroes at `{+-2,+-sqrt(3)}`?

Answer 1

`f(x)=x^4-7x^2+12`

Explanation:

If `f(x)` is a polynomial function with zeros: `{2,-2,sqrt(3),-sqrt(3)}`
then its polynomial has factors
`color(white)("XXX")(x-2), (x+2), (x-sqrt(3))," and " (x+sqrt(3))`

The simplest form of this polynomial will be the product of these (and only these) factors:

`f(x)=underbrace((x-2)(x+2))underbrace((x-sqrt(3))(x+sqrt(3)))`
`color(white)(f(x))=color(white)("xx")(x^2-4) color(white)("xx")*color(white)("xx") (x^2-3)`
`color(white)(f(x))=color(white)("xxxxxx")x^4-7x^2+12`