If r and s are real numbers such that r^3 + s^3 = 0 and r + s = 6, what is the value of 1/r + 1/s?

Introduction / Context:
This problem combines symmetric expressions in algebra, specifically sums and cubes of two real numbers r and s. Questions like this often appear in aptitude tests to check comfort with algebraic identities and the ability to express one symmetric combination in terms of others.

Given Data / Assumptions:

• r and s are real numbers.
• r^3 + s^3 = 0.
• r + s = 6.
• We need to compute 1/r + 1/s.
• We assume r and s are nonzero so that 1/r and 1/s are defined.

Concept / Approach:
We use the identity for the sum of cubes: r^3 + s^3 = (r + s)^3 - 3rs(r + s). This connects r^3 + s^3 with r + s and rs. Once we find rs, we can compute 1/r + 1/s using the identity 1/r + 1/s = (r + s) / (rs).

Step-by-Step Solution:
Step 1: Start from r^3 + s^3 = 0.Step 2: Apply the identity r^3 + s^3 = (r + s)^3 - 3rs(r + s).Step 3: Substitute r + s = 6 into the identity.Step 4: Compute (r + s)^3 = 6^3 = 216.Step 5: So r^3 + s^3 = 216 - 3rs * 6 = 216 - 18rs.Step 6: Given r^3 + s^3 = 0, set 216 - 18rs = 0.Step 7: Solve for rs: 18rs = 216, so rs = 216 / 18 = 12.Step 8: Now compute 1/r + 1/s = (r + s) / (rs) = 6 / 12 = 1/2 = 0.5.

Verification / Alternative check:
We can construct explicit values for r and s. Suppose r + s = 6 and rs = 12, then r and s are roots of t^2 - 6t + 12 = 0. These roots are complex conjugates, but the expression 1/r + 1/s only depends on the symmetric data r + s and rs, so the result 0.5 is still valid and consistent with the algebraic relationships.

Why Other Options Are Wrong:

• 0: This would imply r + s = 0 or rs is infinite, which contradicts r + s = 6.
• 1: This would require rs = 6, but we found rs = 12.
• 2: This would require rs = 3, again inconsistent with the derived value rs = 12.

Common Pitfalls:
A common mistake is to try to solve directly for r and s instead of using symmetric identities, which may become messy. Another frequent error is misapplying the sum of cubes formula or forgetting the factor 3rs(r + s), leading to an incorrect value for rs and hence for 1/r + 1/s.

Final Answer:
The required value of 1/r + 1/s is 0.5.