Question

How do you determine an equation of a polynomial function with zeros at x=2,-2,1 and y-intercept of 24?

Answer 1

`y=x^4+5x^3-10x^2-20x+24`

Explanation:

If `{2,-2,1}` are all zeros of the polynomial
then
the polynomial must contain the factors:
`color(white)("XXX")(x-2)(x+2)(x-1)`

So
`color(white)("XXX")y=(x-2)(x+2)(x-1)xxa` for some additional factor `a`

`bary = (x-2)(x+2)(x-1)=x^3-x^2-4x+4`
which is equal to `4` when `x=0`
So if `y=baryxxa` is to be equal to `24` when `x=0`
then `a` must have the value `6` when `x=0`

The most obvious (but not only) value for `a` is
`color(white)("XXX")(x+6)`
in which case
`color(white)("XXX")y=(x-2)(x+2)(x-1)(x+6)`
`color(white)("XXXX")=x^4+5x^3-10x^2-20x+24`