If p(x) is a cubic polynomial such that p(0) = 0, P(2) = -4 and P(x) is positive only when x > 4 find p(x).

Answer 1

since p(0) = 0
the curve touches at (0,0) but can't cross above the x-axis
which tells me that there must be a double root of 0

since p(x) > 0 only for x>4, it must have crossed the x-axis at 4
so x-4 must be a factor

Let the equation have the form
p(x) = ax^2(x-4)
but (2,-4) lies on it
-4 = 4a(-2)
-4 = -8a
a = 1/2

p(x) = (1/2)x^2 (x-4)

http://www.wolframalpha.com/input/?i=%281%2F2%29x%5E2+%28x-4%29