If 1/p + 1/q = 1/(p + q) for non zero p and q, then what is the value of p^3 − q^3?

Introduction / Context:
This algebra problem deals with a relation between p and q involving reciprocals and asks for p^3 − q^3. Instead of solving explicitly for p and q, we use algebraic manipulation to obtain a condition that implies p^3 and q^3 are equal, so their difference is zero.

Given Data / Assumptions:

• p and q are non zero real numbers.
• 1/p + 1/q = 1/(p + q).
• We must find p^3 − q^3.

Concept / Approach:
We start by simplifying the given fraction identity to derive an algebraic relation between p and q. Then we use the factorization formula p^3 − q^3 = (p − q)(p^2 + pq + q^2). If we can show that p^2 + pq + q^2 = 0, or that p = q, then p^3 − q^3 will be zero.

Step-by-Step Solution:
Step 1: Begin with 1/p + 1/q = 1/(p + q). Step 2: Combine the left side: 1/p + 1/q = (q + p)/(pq). Step 3: Thus (p + q)/(pq) = 1/(p + q). Step 4: Cross multiply (p and q are non zero, and p + q must also be non zero for the right side to be defined): (p + q)^2 = pq. Step 5: Expand the left side: (p + q)^2 = p^2 + 2pq + q^2. Step 6: Set this equal to pq: p^2 + 2pq + q^2 = pq. Step 7: Bring all terms to one side: p^2 + 2pq + q^2 − pq = 0, so p^2 + pq + q^2 = 0. Step 8: Use p^3 − q^3 = (p − q)(p^2 + pq + q^2). Since p^2 + pq + q^2 = 0, we get p^3 − q^3 = (p − q) × 0 = 0.

Verification / Alternative check:
The condition p^2 + pq + q^2 = 0 implies that p and q satisfy a quadratic relation. This has no non zero real solutions, but the algebraic identity still holds symbolically, and p^3 − q^3 evaluates to zero under this relation. The question is algebraic in nature and asks for the value implied by the given condition, which is unambiguously zero.

Why Other Options Are Wrong:
Option A (p − q) and Option B (pq) have incorrect dimensions relative to the derived factorization. Option C (1) has no support from the algebraic manipulation. Option E (p + q) also does not arise from the identity p^3 − q^3 = (p − q)(p^2 + pq + q^2). Only zero is consistent with p^2 + pq + q^2 = 0.

Common Pitfalls:
One common mistake is to incorrectly cross multiply and get (p + q) = pq instead of (p + q)^2 = pq. Another pitfall is forgetting to expand (p + q)^2 properly, leading to sign or coefficient errors. Some students also forget the factorization of p^3 − q^3 and try to cube p and q directly, which is unnecessary and error prone.

Final Answer:

The value of p^3 − q^3 under the given condition is 0.