If x = (a / b) + (b / a), y = (b / c) + (c / b) and z = (c / a) + (a / c), then what is the value of xyz - x^2 - y^2 - z^2?

Introduction / Context:
This problem is an example of an algebraic identity involving symmetric expressions. The variables x, y and z are defined in terms of ratios of three nonzero real numbers a, b and c. You are asked to compute the quantity xyz - x^2 - y^2 - z^2. Although the expression looks complicated, it simplifies to a constant independent of a, b and c, illustrating the power of algebraic manipulation and pattern recognition.

Given Data / Assumptions:

• x = (a / b) + (b / a).
• y = (b / c) + (c / b).
• z = (c / a) + (a / c).
• a, b and c are nonzero real numbers.
• We must evaluate E = xyz - x^2 - y^2 - z^2.

Concept / Approach:
A useful approach is to note that expressions like (a / b) + (b / a) can be written as (a^2 + b^2) / (ab). When you multiply such terms, many factors of a, b and c will combine or cancel. Another strategy is to test with simple numeric values of a, b and c to guess the constant value, and then verify algebraically that the expression does not depend on the specific choice. Because the expression is symmetric, choosing convenient values such as a = b = c can dramatically simplify the computation.

Step-by-Step Solution:
Step 1: Choose simple values that satisfy the assumptions, for example a = b = c = 1. Step 2: Compute x = (1 / 1) + (1 / 1) = 1 + 1 = 2. Step 3: Compute y = (1 / 1) + (1 / 1) = 2 and z = (1 / 1) + (1 / 1) = 2. Step 4: With x = y = z = 2, compute xyz = 2 × 2 × 2 = 8. Step 5: Compute x^2 + y^2 + z^2 = 2^2 + 2^2 + 2^2 = 4 + 4 + 4 = 12. Step 6: Evaluate E = xyz - x^2 - y^2 - z^2 = 8 - 12 = -4.

Verification / Alternative check:
To be sure the value does not depend on the specific choice of a, b and c, try another simple set, such as a = 1, b = 2, c = 3. Compute x, y and z with these values and evaluate E again. You will find that E still equals -4. This strong evidence suggests that the expression is an identity equal to -4 for all nonzero a, b and c. A full algebraic verification would involve expressing each variable as a sum of fractions and carefully expanding, but the symmetry and repeated numeric confirmations strongly support the constant value.

Why Other Options Are Wrong:

• Option b (2) and option c (-1) may arise from incomplete simplification or from evaluating only part of the expression.
• Option d (-6) could result from miscomputing one of the squared terms or miscalculating xyz.
• Option e (0) might seem plausible if someone incorrectly assumes cancellations are complete, but numerical checks contradict this.

Common Pitfalls:
The main risk is attempting a full expansion without recognising symmetry or choosing convenient test values. This can lead to very long expressions and numerous algebraic mistakes. Another pitfall is assuming the result depends on a, b and c and trying to solve for these variables, which is unnecessary. Instead, take advantage of the freedom to assign simple values within the constraints, verify the result with more than one example, and then recognise that the expression is constant.

Final Answer:
The value of xyz - x^2 - y^2 - z^2 is -4.