In aptitude (symmetric polynomial identities), if (x + y + z) = 12, (xy + yz + zx) = 44, and xyz = 48, use the standard identity for the sum of cubes to find the value of x^3 + y^3 + z^3.

Introduction / Context:
This question focuses on symmetric polynomials in three variables and the use of algebraic identities relating sums, pairwise products, and products of variables to the sum of their cubes. Instead of finding x, y, and z individually, we use the identity for x^3 + y^3 + z^3 in terms of x + y + z, xy + yz + zx, and xyz. Problems of this kind are popular in higher level aptitude and algebra exams, as they test conceptual understanding of identities and symbolic manipulation.

Given Data / Assumptions:
- x + y + z = 12.
- xy + yz + zx = 44.
- xyz = 48.
- We need to find x^3 + y^3 + z^3.
- All variables are real numbers, but their exact values are not required.

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 already have x + y + z, xy + yz + zx, and xyz. To use the identity, we also need x^2 + y^2 + z^2, which can be found using another identity:
(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx).
Combining these identities allows us to find x^3 + y^3 + z^3 directly.

Step-by-Step Solution:
Step 1: Use the square identity to find x^2 + y^2 + z^2.(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx).Step 2: Substitute the known values: 12^2 = x^2 + y^2 + z^2 + 2 * 44.Step 3: Compute 12^2 = 144 and 2 * 44 = 88.Step 4: So 144 = x^2 + y^2 + z^2 + 88, which gives x^2 + y^2 + z^2 = 144 - 88 = 56.Step 5: Now use the sum of cubes identity: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx).Step 6: Compute the expression inside the parentheses: x^2 + y^2 + z^2 - xy - yz - zx = 56 - 44 = 12.Step 7: So the right-hand side becomes (x + y + z) * 12 = 12 * 12 = 144.Step 8: Therefore, x^3 + y^3 + z^3 - 3xyz = 144.Step 9: Substitute xyz = 48: x^3 + y^3 + z^3 - 3 * 48 = 144.Step 10: Compute 3 * 48 = 144.Step 11: So x^3 + y^3 + z^3 - 144 = 144, which gives x^3 + y^3 + z^3 = 288.

Verification / Alternative check:
We could attempt to construct a cubic polynomial with roots x, y, z using the given symmetric sums. The polynomial would be t^3 - (x + y + z)t^2 + (xy + yz + zx)t - xyz = 0, that is t^3 - 12t^2 + 44t - 48 = 0. If we could factor this polynomial and find explicit roots, we could then cube and sum them to verify 288, but this is more time consuming. Since the symmetric identities are well established and the calculations here are straightforward, the identity based method is both efficient and reliable.

Why Other Options Are Wrong:
- 104, 144, 196, 240: These values do not satisfy the identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) when we plug in the given data. They are likely results of arithmetic mistakes, such as miscomputing x^2 + y^2 + z^2 or forgetting to multiply xyz by 3.

Common Pitfalls:
Students often misremember the sum of cubes identity, sometimes omitting the -3xyz term or misplacing signs. Another common error is computing (x + y + z)^2 incorrectly or forgetting to subtract 2(xy + yz + zx) when solving for x^2 + y^2 + z^2. Careful substitution and stepwise calculation are essential. Also, trying to find x, y, z individually is usually unnecessary and wastes time in exam conditions.

Final Answer:
The value of the sum of cubes is x^3 + y^3 + z^3 = 288.