If x, y and z are real numbers such that x^3 + y^3 + z^3 = 13, x + y + z = 1 and xyz = 1, then what is the value of xy + yz + zx?

Introduction / Context:
This algebra question uses symmetric identities involving the sum of cubes of three numbers, their sum and their product. We are asked to find the sum of pairwise products xy + yz + zx, which is a standard symmetric expression. Such problems are common in algebraic identities and polynomial root based questions.

Given Data / Assumptions:

• x, y, z are real numbers.
• x^3 + y^3 + z^3 = 13.
• x + y + z = 1.
• xyz = 1.
• We must find S = xy + yz + zx.

Concept / Approach:
The key identity is: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). We will use the given values to convert this identity into an equation involving S. We also use the identity: (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx). These two together allow us to solve for S.

Step-by-Step Solution:
Let S1 = x + y + z and S2 = xy + yz + zx. Given S1 = 1 and xyz = 1. Use the cubes identity: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). Substitute given values: 13 - 3*1 = 10. So 10 = S1(x^2 + y^2 + z^2 - S2) = 1*(x^2 + y^2 + z^2 - S2). Hence x^2 + y^2 + z^2 - S2 = 10, so x^2 + y^2 + z^2 = 10 + S2. Now use (x + y + z)^2 = x^2 + y^2 + z^2 + 2S2. S1^2 = 1^2 = 1 = (10 + S2) + 2S2 = 10 + 3S2. Thus 3S2 = 1 - 10 = -9. So S2 = -9 / 3 = -3. Therefore xy + yz + zx = -3.

Verification / Alternative check:
We can verify the logic by observing that the calculation depends only on symmetric combinations of x, y, z. Any triple of numbers satisfying the given conditions will produce the same S2. There is no need to find x, y and z individually, which would be complicated.

Why Other Options Are Wrong:
Values -1, 1, 0 and 3 would appear if the identities were misapplied or if 3xyz was forgotten in the sum of cubes formula. For instance, setting 13 equal directly to (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) would produce a different incorrect value.

Common Pitfalls:
Students often forget the full form of the identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). Another pitfall is mishandling the square (x + y + z)^2, especially the middle term 2(xy + yz + zx). Keeping track of each symbol using S1, S2 and xyz helps to avoid confusion.

Final Answer:
The value of xy + yz + zx is -3.