Find a cubic polynomial if p(0)=0 and p(2)= -4 and p(x) > 0 only if x > 4, find the cubic polynomial.

Answer 1

p(x) = (x)* (x-a)(x-b)
if x = 2, p = -4

p(2) = -4 = (2)(2-a)(2-b)
-2 = 4 -2a - 2b + ab
-6 = -2a -2b + ab

p(4) = 0
0 = (4)(4-a)(4-b)
-16 = - 4a -4b + ab

10 = 2a +2b
5 = a + b
b = 5-a

p(x) = (x)* (x-a)(x-b)
p(x) = x * (x-a)(x+a-5)
if x = 2
-4 = 2*(2-a)(-3+a)
-2 = -6 +5a -a^2
a^2 - 5 a +4 = 0
(a-4)(a-1) = 0
a = 4 or a = 1
b = 1 or b = 4

p(x) = (x)* (x-a)(x-b)
using a = 4, b = 1
p = x(x-4)(x-1)
does that work - check it :)