<p>The polynomial \(f(x) = x^4 + ax^3 + bx^2 + cx + d\) has real coefficients and \(f(2i) = f(2+i) = 0\). Find the value of \((a + b + c + d)\).</p>
Step-by-Step Solution
Key Concept: Since f(x) has real coefficients, complex roots must appear in conjugate pairs: if 2i is a root, then -2i is also a root; if 2+i is a root, then 2-i is also a root. These four roots completely determine the quartic polynomial.
<p><strong>Step 1:</strong> Identify all roots using the conjugate pair property. Since f(x) has real coefficients:</p><ul><li>If 2i is a root, then -2i is also a root</li><li>If 2+i is a root, then 2-i is also a root</li></ul><p><strong>Step 2:</strong> Write f(x) as a product of factors:</p><p>f(x) = (x - 2i)(x + 2i)(x - (2+i))(x - (2-i))</p><p><strong>Step 3:</strong> Simplify the conjugate pairs:</p><p>(x - 2i)(x + 2i) = x² - (2i)² = x² - (-4) = x² + 4</p><p>(x - (2+i))(x - (2-i)) = ((x-2) - i)((x-2) + i) = (x-2)² - i² = (x-2)² + 1 = x² - 4x + 4 + 1 = x² - 4x + 5</p><p><strong>Step 4:</strong> Multiply the two quadratics:</p><p>f(x) = (x² + 4)(x² - 4x + 5)</p><p>= x⁴ - 4x³ + 5x² + 4x² - 16x + 20</p><p>= x⁴ - 4x³ + 9x² - 16x + 20</p><p><strong>Step 5:</strong> Identify coefficients: a = -4, b = 9, c = -16, d = 20</p><p><strong>Step 6:</strong> Calculate a + b + c + d:</p><p>a + b + c + d = -4 + 9 + (-16) + 20 = 9</p><p>∴ Answer: <strong>9</strong></p>
Correct Answer: 9