If x + y + z = 1, 1/x + 1/y + 1/z = 1, and xyz = -1 for non zero real numbers x, y and z, then what is the value of the sum of cubes x^3 + y^3 + z^3?

Introduction / Context:
This problem is a classic example of using symmetric identities for three variables. We are given the sum, the sum of reciprocals, and the product of x, y, and z, and we are asked to find x^3 + y^3 + z^3. Such questions appear often in algebra sections of competitive examinations.

Given Data / Assumptions:

• x, y, z are non zero real numbers.
• x + y + z = 1.
• 1/x + 1/y + 1/z = 1.
• xyz = -1.
• We must find x^3 + y^3 + z^3.

Concept / Approach:
We use two main ideas:

• The relationship between 1/x + 1/y + 1/z and xy + yz + zx.
• The identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)^3 - 3(x + y + z)(xy + yz + zx).

Once we find xy + yz + zx, we can substitute everything into the cubic identity and simplify.

Step-by-Step Solution:
Step 1: Use the reciprocal relation: 1/x + 1/y + 1/z = (xy + yz + zx)/(xyz). Step 2: Substitute xyz = -1 and the given value 1/x + 1/y + 1/z = 1. Step 3: We get (xy + yz + zx)/(-1) = 1, so xy + yz + zx = -1. Step 4: Use the identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)^3 - 3(x + y + z)(xy + yz + zx). Step 5: Substitute x + y + z = 1, xy + yz + zx = -1, and xyz = -1. Step 6: Compute the right side: (1)^3 - 3(1)(-1) = 1 + 3 = 4. Step 7: Therefore x^3 + y^3 + z^3 - 3(-1) = 4, which gives x^3 + y^3 + z^3 + 3 = 4. Step 8: Hence x^3 + y^3 + z^3 = 1.

Verification / Alternative check:
The symmetry suggests that actual numeric solutions exist, but solving directly for x, y, and z is complicated. Instead, checking the algebraic identity is more reliable. Every step uses standard symmetric identities, and the operations are reversible, so the final result is consistent with the given conditions.

Why Other Options Are Wrong:
Option A (-1) or Option C (-2) would arise only if signs in the identity were mishandled, especially around -3xyz. Option D (2) corresponds to using (x + y + z)^3 alone and ignoring the -3(x + y + z)(xy + yz + zx) term. Option E (0) has no algebraic support from the given relationships.

Common Pitfalls:
Typical mistakes include confusing the identity for two variables with the three variable identity, or forgetting that 1/x + 1/y + 1/z equals (xy + yz + zx)/(xyz). Sign errors around xyz = -1 are also very common. Careful substitution and stepwise simplification are essential in such problems.

Final Answer:

The required sum of cubes is 1.