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

##### 1 Answer

#### 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

As a result, it satisfies the required conditions.

Any polynomial in

Any positive scalar multiple of

If

#{ (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

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