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Solve for a^3 from a reciprocal linear relation: If a + 1/a = 1 (a ≠ 0), determine the value of a^3.

Difficulty: Medium

Correct Answer: – 1

Explanation:


Introduction / Context:
The relation a + 1/a = 1 leads to a quadratic equation in a. From that, powers of a can be reduced systematically. Here we are asked for a^3, which can be found without computing any decimals by using the quadratic to eliminate higher powers.


Given Data / Assumptions:

  • a + 1/a = 1
  • a ≠ 0
  • Find a^3


Concept / Approach:
Multiply the given equation by a to form a quadratic: a^2 − a + 1 = 0. Then express a^2 in terms of a and substitute into a^3 = a*a^2 to reduce a^3 to a linear expression in a, which then collapses to a constant using the same quadratic again.


Step-by-Step Solution:

From a + 1/a = 1 ⇒ multiply by a: a^2 + 1 = a ⇒ a^2 − a + 1 = 0.Hence a^2 = a − 1.Compute a^3 = a * a^2 = a(a − 1) = a^2 − a.Replace a^2 using a^2 − a + 1 = 0 ⇒ a^2 = a − 1.Thus a^3 = (a − 1) − a = −1.


Verification / Alternative check:

The roots of a^2 − a + 1 = 0 are complex conjugates with magnitude 1; their cube equals −1, consistent with the algebraic reduction.


Why Other Options Are Wrong:

  • 2, 4, −2: None satisfy the derived identity; substituting back into a + 1/a = 1 fails.


Common Pitfalls:

  • Mistaking a^2 − a + 1 = 0 for the cube roots of unity equation a^2 + a + 1 = 0.
  • Arithmetic slips when computing a^3 = a(a − 1).


Final Answer:

– 1
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