Question

How do you define a polynomial `p(x)` that is zero or negative whenever `x in [-2, 1] uu [4, oo)` ?

Answer 1

`p(x) = -(x+2)(x-1)(x-4) = -x^3+3x^2+6x-8`

Explanation:

Define:

`p(x) = -(x+2)(x-1)(x-4) = -x^3+3x^2+6x-8`

graph{ -x^3+3x^2+6x-8 [-10, 10, -15, 15]}

This cubic has a leading negative coefficient `-1` and zeros at `x=-2`, `x=1` and `x=4`.

As a result, it satisfies the required conditions.

Any polynomial in `x` with these zeros will be a multiple (scalar or polynomial) of `p(x)`.

Any positive scalar multiple of `p(x)` will also be a cubic satisfying the conditions.

If `p(x)` is multiplied by any polynomial `q(x)` satisfying the following conditions, then the resulting polynomial also satisfies the conditions:

`{ (q(x) > 0 " for all " x in (-oo, -2) uu (-2, 1) uu (1, 4) uu (4, oo)), (q(x) >= 0 " for all " x in { -2, 1, 4 }) :}`

For example, we can multiply `p(x)` by `q(x) = (x-1)^2(x^2+2)` to get a septic polynomial that is non-positive in `[-2, 1] uu [4, oo)`.

graph{(-x^3+3x^2+6x-8)(x-1)^2(x^2+2) [-10, 10, -240, 700]}