p(k)=k(k-1)(k+1)

Solution for p(k)=k(k-1)(k+1) equation:

Simplifying
p(k) = k(k + -1)(k + 1)

Multiply p * k
kp = k(k + -1)(k + 1)

Reorder the terms:
kp = k(-1 + k)(k + 1)

Reorder the terms:
kp = k(-1 + k)(1 + k)

Multiply (-1 + k) * (1 + k)
kp = k(-1(1 + k) + k(1 + k))
kp = k((1 * -1 + k * -1) + k(1 + k))
kp = k((-1 + -1k) + k(1 + k))
kp = k(-1 + -1k + (1 * k + k * k))
kp = k(-1 + -1k + (1k + k2))

Combine like terms: -1k + 1k = 0
kp = k(-1 + 0 + k2)
kp = k(-1 + k2)
kp = (-1 * k + k2 * k)
kp = (-1k + k3)

Solving
kp = -1k + k3

Solving for variable 'k'.

Reorder the terms:
k + kp + -1k3 = -1k + k3 + k + -1k3

Reorder the terms:
k + kp + -1k3 = -1k + k + k3 + -1k3

Combine like terms: -1k + k = 0
k + kp + -1k3 = 0 + k3 + -1k3
k + kp + -1k3 = k3 + -1k3

Combine like terms: k3 + -1k3 = 0
k + kp + -1k3 = 0

Factor out the Greatest Common Factor (GCF), 'k'.
k(1 + p + -1k2) = 0

Subproblem 1
Set the factor 'k' equal to zero and attempt to solve:

Simplifying
k = 0

Solving
k = 0

Move all terms containing k to the left, all other terms to the right.

Simplifying
k = 0
Subproblem 2
Set the factor '(1 + p + -1k2)' equal to zero and attempt to solve:

Simplifying
1 + p + -1k2 = 0

Reorder the terms:
1 + -1k2 + p = 0

Solving
1 + -1k2 + p = 0

Move all terms containing k to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1k2 + -1 + p = 0 + -1

Reorder the terms:
1 + -1 + -1k2 + p = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -1k2 + p = 0 + -1
-1k2 + p = 0 + -1

Combine like terms: 0 + -1 = -1
-1k2 + p = -1

Add '-1p' to each side of the equation.
-1k2 + p + -1p = -1 + -1p

Combine like terms: p + -1p = 0
-1k2 + 0 = -1 + -1p
-1k2 = -1 + -1p

Divide each side by '-1'.
k2 = 1 + p

Simplifying
k2 = 1 + p

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
k = {0}