Find a cubic polynomial p(x) so that the solution of p(x) ≤ 0 is [-2,1] or [4,∞). Is your polynomial the only one that gives this solution? Explain.

Answer 1

clearly you need roots at -2,1,4

Since p < 0 on the first interval, it must be >0 when x < -2.

So, the leading coefficient must be negative, giving us

p(x) = -a(x+2)(x-1)(x-4)
where a is any positive value.

See the graph for a=1 here:

http://www.wolframalpha.com/input/?i=-1(x%2B2)(x-1)(x-4)