(1+2z^1-1z^3)=1

Solution for (1+2z^1-1z^3)=1 equation:

Simplifying
(1 + 2z + -1z3) = 1

Remove parenthesis around (1 + 2z + -1z3)
1 + 2z + -1z3 = 1

Add '-1' to each side of the equation.
1 + 2z + -1 + -1z3 = 1 + -1

Reorder the terms:
1 + -1 + 2z + -1z3 = 1 + -1

Combine like terms: 1 + -1 = 0
0 + 2z + -1z3 = 1 + -1
2z + -1z3 = 1 + -1

Combine like terms: 1 + -1 = 0
2z + -1z3 = 0

Solving
2z + -1z3 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z'.
z(2 + -1z2) = 0

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

Simplifying
z = 0

Solving
z = 0

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

Simplifying
z = 0
Subproblem 2
Set the factor '(2 + -1z2)' equal to zero and attempt to solve:

Simplifying
2 + -1z2 = 0

Solving
2 + -1z2 = 0

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

Add '-2' to each side of the equation.
2 + -2 + -1z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -1z2 = 0 + -2
-1z2 = 0 + -2

Combine like terms: 0 + -2 = -2
-1z2 = -2

Divide each side by '-1'.
z2 = 2

Simplifying
z2 = 2

Take the square root of each side:
z = {-1.414213562, 1.414213562}
Solution
z = {0, -1.414213562, 1.414213562}