Quadratic Equations
Roots and polynomial expressions
Grade 11
Question:
<p>If \(\alpha\) is the root of the equation \(x^2 - x + 2 = 0\), then the value of \(\dfrac{6(-\alpha^3 + 2\alpha^2 - \alpha)}{\alpha^5 - 3\alpha^4 + 3\alpha^3 - \alpha^2}\) is equal to:</p>
<p>3</p>
<p>6</p>
<p>9</p>
<p>12</p>
Step-by-Step Solution
Key Concept: Since α is a root of x² - x + 2 = 0, we have α² = α - 2. Use this relation to reduce all higher powers of α to linear expressions, then simplify the given fraction.
<p><strong>Step 1: Use the root property</strong></p><p>Since α satisfies x² - x + 2 = 0, we have: <strong>α² = α - 2</strong></p><p><strong>Step 2: Reduce higher powers</strong></p><p>α³ = α · α² = α(α - 2) = α² - 2α = (α - 2) - 2α = -α - 2</p><p>α⁴ = α · α³ = α(-α - 2) = -α² - 2α = -(α - 2) - 2α = -3α + 2</p><p>α⁵ = α · α⁴ = α(-3α + 2) = -3α² + 2α = -3(α - 2) + 2α = -α + 6</p><p><strong>Step 3: Simplify numerator</strong></p><p>6(-α³ + 2α² - α) = 6(-(-α - 2) + 2(α - 2) - α)</p><p>= 6(α + 2 + 2α - 4 - α) = 6(2α - 2) = 12(α - 1)</p><p><strong>Step 4: Simplify denominator</strong></p><p>α⁵ - 3α⁴ + 3α³ - α² = (-α + 6) - 3(-3α + 2) + 3(-α - 2) - (α - 2)</p><p>= -α + 6 + 9α - 6 - 3α - 6 - α + 2</p><p>= 4α - 4 = 4(α - 1)</p><p><strong>Step 5: Final division</strong></p><p>$$ rac{12(α - 1)}{4(α - 1)} = rac{12}{4} = 3$$</p><p>∴ Answer: <strong>B</strong></p>
Correct Answer: B