If x + y + z = 22 and xy + yz + zx = 35, what is the value of (x - y)^2 + (y - z)^2 + (z - x)^2?

Introduction / Context:
This algebra problem asks you to compute a symmetric expression involving squared differences of three variables x, y and z. Instead of solving for x, y and z individually, we are given the sum x + y + z and the sum of pairwise products xy + yz + zx. The goal is to use identities to express (x - y)^2 + (y - z)^2 + (z - x)^2 in terms of these known quantities. This tests understanding of symmetric polynomials and the ability to manipulate algebraic identities efficiently.

Given Data / Assumptions:

• x + y + z = 22.
• xy + yz + zx = 35.
• We need S = (x - y)^2 + (y - z)^2 + (z - x)^2.
• No explicit values of x, y and z are required; we use identities.

Concept / Approach:
There is a well known identity for the sum of squared differences: (x - y)^2 + (y - z)^2 + (z - x)^2 = 2(x^2 + y^2 + z^2 - xy - yz - zx). Therefore, our plan is to find x^2 + y^2 + z^2 using the given information and then substitute into this identity. We can use the square of the sum (x + y + z)^2 to relate x^2 + y^2 + z^2 with xy + yz + zx.

Step-by-Step Solution:
Step 1: Recall the identity for the square of a sum: (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx). Step 2: Substitute the given values: (x + y + z) = 22, so (x + y + z)^2 = 22^2 = 484. Step 3: Substitute xy + yz + zx = 35 into the identity: 484 = x^2 + y^2 + z^2 + 2 * 35 = x^2 + y^2 + z^2 + 70. Step 4: Solve for x^2 + y^2 + z^2: x^2 + y^2 + z^2 = 484 - 70 = 414. Step 5: Use the identity for the sum of squared differences: S = (x - y)^2 + (y - z)^2 + (z - x)^2 = 2(x^2 + y^2 + z^2 - xy - yz - zx). Step 6: Substitute the known values: S = 2(414 - 35) = 2 * 379 = 758.

Verification / Alternative check:
To verify, one could attempt to find specific numbers x, y and z that satisfy x + y + z = 22 and xy + yz + zx = 35 and then compute the squared differences directly. However, finding such values systematically is more complex than using the identity. Since both identities used are standard and we applied them correctly, and since the final arithmetic 2 * 379 = 758 is straightforward, the result is reliable.

Why Other Options Are Wrong:
The other numeric choices 793, 681, 715 and 700 would arise from arithmetic mistakes such as miscomputing 22^2, forgetting to subtract 70, or misapplying the factor of 2 in the final identity. For example, using 414 - 35 without multiplying by 2 would give 379, or multiplying an incorrect intermediate value by 2 would give a different result. Only 758 matches the correct application of both identities and the correct arithmetic.

Common Pitfalls:
A common error is to try to expand each squared difference (x - y)^2, (y - z)^2 and (z - x)^2 separately and then sum them, which is algebraically correct but more tedious and prone to mistakes. Another issue is forgetting the factor of 2 in the identity for S. Some students also incorrectly use (x + y + z)^2 = x^2 + y^2 + z^2 + xy + yz + zx instead of including the factor 2. Remembering the exact structure of these identities and substituting carefully helps avoid such errors.

Final Answer:
Thus, the value of (x - y)^2 + (y - z)^2 + (z - x)^2 is 758.