For which value of k will the polynomial expression x^6 − 18x^3 + k become a perfect square of a polynomial in x? Choose the correct value of k.

Introduction / Context:
This question checks pattern recognition for perfect-square polynomials. The expression involves x^6 and x^3, suggesting a substitution like t = x^3. Then the polynomial becomes a quadratic in t, which can be matched to a perfect square form (t − a)^2.

Given Data / Assumptions:

• Expression: x^6 − 18x^3 + k • Required: choose k so the expression is a perfect square polynomial • Perfect square means it can be written as (polynomial)^2 exactly

Concept / Approach:
Let t = x^3. Then x^6 = (x^3)^2 = t^2. The expression becomes: t^2 − 18t + k. A quadratic is a perfect square when it matches: (t − a)^2 = t^2 − 2at + a^2. Match coefficients: −2a = −18, so a = 9. Then k must equal a^2 = 81.

Step-by-Step Solution:
1) Substitute t = x^3 2) Rewrite: x^6 − 18x^3 + k = t^2 − 18t + k 3) Perfect square form: (t − a)^2 = t^2 − 2at + a^2 4) Match the coefficient of t: −2a = −18 → a = 9 5) Therefore k = a^2 = 9^2 = 81 6) So the expression becomes: t^2 − 18t + 81 = (t − 9)^2 7) Convert back: (x^3 − 9)^2

Verification / Alternative check:
Expand (x^3 − 9)^2: = x^6 − 2*9*x^3 + 81 = x^6 − 18x^3 + 81. This matches exactly, confirming k = 81.

Why Other Options Are Wrong:
• 9 or 27: too small; does not match a^2 when a = 9. • −9 or −81: a perfect square polynomial cannot have a negative constant term here if it is (x^3 − 9)^2.

Common Pitfalls:
• Forgetting to treat x^6 as (x^3)^2. • Using (t + a)^2 instead of (t − a)^2, causing sign mismatch.

Final Answer:
81