p^3 + q^3 + r^3 – 3pqr = 4. If a = q + r, b = r + p and c = p + q, then what is the value of a^3 + b^3 + c^3 – 3abc?

Introduction / Context:
This question is about symmetric expressions in three variables and uses a known factorization for sums of cubes. The given relation involves p^3, q^3, r^3, and pqr, and we are asked to find a related expression in a, b, and c where each of these variables is a sum of two of p, q, and r.

Given Data / Assumptions:

• p^3 + q^3 + r^3 - 3pqr = 4.
• a = q + r, b = r + p, c = p + q.
• We must find the value of a^3 + b^3 + c^3 - 3abc.

Concept / Approach:
We recall the identity p^3 + q^3 + r^3 - 3pqr = (p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp). A similar identity holds for a, b, and c. The strategy is to express a^3 + b^3 + c^3 - 3abc in terms of p, q, and r, then compare it with the original expression. In fact, the new expression turns out to be exactly twice the original value.

Step-by-Step Solution:
Start with a = q + r, b = r + p, c = p + q. We want E = a^3 + b^3 + c^3 - 3abc. Expand E using known factorization. For three variables, we have identity: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). But instead of expanding fully in a, b, and c, it is more efficient to express E directly in terms of p, q, and r. After algebraic manipulation one obtains: E = 2(p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp). Notice that p^3 + q^3 + r^3 - 3pqr can itself be factored as: p^3 + q^3 + r^3 - 3pqr = (p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp). Therefore E equals 2 times this original expression: E = 2 (p^3 + q^3 + r^3 - 3pqr). The problem states that p^3 + q^3 + r^3 - 3pqr = 4. Thus E = 2 × 4 = 8.

Verification / Alternative check:
We can verify this relationship by choosing specific numbers p, q, r that satisfy p^3 + q^3 + r^3 - 3pqr = 4 and checking that the corresponding a, b, and c give E = 8. Symbolic algebra or careful numerical substitution confirms that the identity E = 2(p^3 + q^3 + r^3 - 3pqr) holds for all p, q, and r.

Why Other Options Are Wrong:
4 and 2: These correspond to using the original value directly or half of it, which would be correct only if E were equal to the base expression or half of it. We have shown it is exactly double.
6 and 12: These suggest multiplying 4 by three halves or by three, which does not match the factorization pattern.

Common Pitfalls:
A common difficulty is trying to expand a^3, b^3, c^3 fully and then simplify, which can be very long and error prone. Another mistake is not recognizing the structural similarity between the expressions in p, q, r and in a, b, c. Using known identities for sums of cubes and factoring early makes the problem short and elegant.

Final Answer:
The value of a^3 + b^3 + c^3 - 3abc is 8.