Quadratic Equations
Roots of unity
Grade 11
Question:
<p>If roots of an equation \(x^n - 1 = 0\) are \(1, a_1, a_2, \ldots, a_{n-1}\), then the value of \((1 - a_1)(1 - a_2)(1 - a_3)\cdots(1 - a_{n-1})\) will be</p>
<p>(1) \(n\)</p>
<p>(2) \(n^2\)</p>
<p>(3) \(n^n\)</p>
<p>(4) 0</p>
Step-by-Step Solution
Key Concept: The roots of x^n - 1 = 0 are the nth roots of unity. Use the factorization x^n - 1 = (x-1)(x-a₁)(x-a₂)...(x-aₙ₋₁) and evaluate the derivative or use polynomial division to find the product (1-a₁)(1-a₂)...(1-aₙ₋₁).
<p><strong>Step 1:</strong> The equation x^n - 1 = 0 has roots that are the nth roots of unity: 1, a₁, a₂, ..., aₙ₋₁.</p><p><strong>Step 2:</strong> Factor the polynomial: x^n - 1 = (x - 1)(x - a₁)(x - a₂)...(x - aₙ₋₁)</p><p><strong>Step 3:</strong> Divide both sides by (x - 1): (x^n - 1)/(x - 1) = (x - a₁)(x - a₂)...(x - aₙ₋₁)</p><p><strong>Step 4:</strong> Using polynomial division or the geometric series formula: (x^n - 1)/(x - 1) = x^(n-1) + x^(n-2) + ... + x + 1</p><p><strong>Step 5:</strong> Evaluate at x = 1: The left side gives 1 + 1 + ... + 1 (n times) = n. The right side gives (1 - a₁)(1 - a₂)...(1 - aₙ₋₁).</p><p>∴ Answer: <strong>n</strong></p>
Correct Answer: A