<p>If \(\alpha^2 - \alpha + 2 = 0\), then find the value of \(\dfrac{6(-\alpha^3 + 2\alpha^2 - \alpha)}{\alpha^5 - 3\alpha^4 + 3\alpha^3 - \alpha^2}\).</p>
Step-by-Step Solution
Key Concept: Since α satisfies α² - α + 2 = 0, we have α² = α - 2. Use this relation recursively to reduce all higher powers of α to linear expressions, then simplify the given fraction using the constraint.
<p><strong>Step 1:</strong> From α² - α + 2 = 0, we get <strong>α² = α - 2</strong>. This is our key constraint.</p><p><strong>Step 2:</strong> Find α³: α³ = α·α² = α(α - 2) = α² - 2α = (α - 2) - 2α = -α - 2</p><p><strong>Step 3:</strong> Find α⁴: α⁴ = α·α³ = α(-α - 2) = -α² - 2α = -(α - 2) - 2α = -3α + 2</p><p><strong>Step 4:</strong> Find α⁵: α⁵ = α·α⁴ = α(-3α + 2) = -3α² + 2α = -3(α - 2) + 2α = -α + 6</p><p><strong>Step 5:</strong> Simplify the <strong>numerator</strong>: 6(-α³ + 2α² - α) = 6(-(-α - 2) + 2(α - 2) - α) = 6(α + 2 + 2α - 4 - α) = 6(2α - 2) = <strong>12(α - 1)</strong></p><p><strong>Step 6:</strong> Simplify the <strong>denominator</strong>: α⁵ - 3α⁴ + 3α³ - α² = (-α + 6) - 3(-3α + 2) + 3(-α - 2) - (α - 2)</p><p>= -α + 6 + 9α - 6 - 3α - 6 - α + 2 = 4α - 4 = <strong>4(α - 1)</strong></p><p><strong>Step 7:</strong> Final answer: 12(α - 1) / 4(α - 1) = <strong>3</strong></p><p>∴ Answer: A</p>
Correct Answer: A