<p>In the expansion of \((1 + 2x + x^2)^9\) there is exactly one term whose coefficient is not equal to the coefficient of any other term. (State whether true or false.)</p>
Step-by-Step Solution
Key Concept: Rewrite (1 + 2x + x²)⁹ = [(1 + x)²]⁹ = (1 + x)¹⁸, then use the binomial expansion where the coefficient of xʳ is C(18,r). A coefficient appears exactly once if and only if C(18,r) ≠ C(18,s) for all r ≠ s.
<p><strong>Step 1:</strong> Recognize that 1 + 2x + x² = (1 + x)²</p><p><strong>Step 2:</strong> Rewrite the expansion: (1 + 2x + x²)⁹ = [(1 + x)²]⁹ = (1 + x)¹⁸</p><p><strong>Step 3:</strong> Apply binomial theorem: (1 + x)¹⁸ = Σ C(18,r)xʳ for r = 0 to 18</p><p><strong>Step 4:</strong> Identify unique coefficients. The binomial coefficients are: C(18,0), C(18,1), C(18,2), ..., C(18,18)</p><p><strong>Step 5:</strong> Use symmetry property: C(18,r) = C(18,18-r). This means coefficients appear in pairs, EXCEPT when r = 18-r, which gives r = 9</p><p><strong>Step 6:</strong> Therefore C(18,9) is the only coefficient that appears exactly once (the middle term)</p><p><strong>Step 7:</strong> All other coefficients appear at least twice due to symmetry</p><p>∴ Answer: <strong>True</strong> - There is exactly one term (the coefficient C(18,9)) whose coefficient equals no other term's coefficient.</p>
Correct Answer: A