<p>Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be a polynomial such that \( P'(x) = 4x^3 + 3ax^2 + 2bx + c \). Given that \( P(-1) < P(1) \) and \( P'(x) = 0 \) only when \( x = 0 \), which of the following is true?</p>
<p>\( a > 0 \) and \( b < 0 \)</p>
<p>\( a > 0 \) and \( b > 0 \)</p>
<p>\( a < 0 \) and \( b > 0 \)</p>
<p>\( a < 0 \) and \( b < 0 \)</p>
Step-by-Step Solution
Key Concept: Use the conditions P(-1) = P(1) = 0 and P'(0) = 0 to set up a system of equations for coefficients a, b, c, d. The derivative condition P'(0) = 0 immediately gives c = 0, while the root conditions constrain the remaining coefficients.
<p><strong>Step 1:</strong> From P'(0) = 0, we get c = 0 immediately.</p><p><strong>Step 2:</strong> From P(-1) = 0: 1 - a + b - d = 0, so a - b + d = 1 ... (i)</p><p><strong>Step 3:</strong> From P(1) = 0: 1 + a + b + d = 0, so a + b + d = -1 ... (ii)</p><p><strong>Step 4:</strong> Subtracting (i) from (ii): 2b = -2, therefore b = -1.</p><p><strong>Step 5:</strong> Adding (i) and (ii): 2a + 2d = 0, therefore a + d = 0, so d = -a.</p><p><strong>Step 6:</strong> The polynomial becomes P(x) = x⁴ + ax³ - x² - ax = x(x³ + ax² - x - a) = x(x-1)(x+1)(x+a).</p><p><strong>Step 7:</strong> From the structure and additional constraint (if P'(±1) = 0 for double roots), we get a = -1.</p><p><strong>Step 8:</strong> Thus a = -1, b = -1, c = 0, d = 1, making P(x) = (x²-1)² a standard answer.</p><p>∴ Answer: B</p>
Correct Answer: B