Answer:
p(x) is a function, and a polynomial of degree 3, then
By remainder theorem and factor theorem,
p(0) = p(1) = p (-1) = 0 if and only if
x = 0, 1, -1 are zeroes if and only if
p(x) = k(x - 0)(x - 1)(x + 1) = k(x)(x^2 - 1) = k(x^3 - x) = kx^3 - kx
p(2) = k(2)^3 - k(2) = 6
8k - 2k = 6
6k = 6
k = 1
Thus p(x) = x(x - 1)(x + 1) = x(x^2 - 1) = x^3 - x
p(-x) = (-x)^3 - (-x) = x^3 + x
-p(x) = -(x^3 - x) = -x^3 + x
As you can see, p(x) does not equal to -p(x), unless x = 0, 1, -1,
because p(0) = -p(0) = 0, p(-1) = - p(1) = 0, p(1) = -p(-1) = 0
p(x) = x(x - 1)(x + 1)
If x < -1, then x < 0, x - 1 < 0, x + 1 < 0
That means p(x) < 0
If -1 < x < 0, then x < 0, -2 < x - 1 < -1 < 0, 0 < x + 1 < 1
That means p(x) > 0
If 0 < x < 1, then x > 0, -1 < x - 1 < 0, 0 < 1 < x + 1 < 2
That means p(x) < 0
If x > 1, then x > 0, x - 1 > 0, x + 1 > 2 > 0
That means p(x) > 0
Thus p(x) < 0 if x < -1 or 0 < x < 1