<p>\(P(x)\) is a polynomial with integral coefficients such that for four distinct integers \(a, b, c, d\), \(P(a) = P(b) = P(c) = P(d) = 3\). If \(P(e) = 5\) (\(e\) is an integer), then</p>
Step-by-Step Solution
Key Concept: If P(x) - 3 has four distinct integer roots a, b, c, d, then P(x) - 3 = (x-a)(x-b)(x-c)(x-d)Q(x) for some polynomial Q(x). Since P(e) = 5, we have (e-a)(e-b)(e-c)(e-d)Q(e) = 2, which severely constrains the possible values.
<p><strong>Step 1:</strong> Since P(a) = P(b) = P(c) = P(d) = 3, the polynomial P(x) - 3 has roots at x = a, b, c, d. Therefore:</p><p>P(x) - 3 = (x-a)(x-b)(x-c)(x-d)·Q(x)</p><p>where Q(x) is a polynomial with integral coefficients.</p><p><strong>Step 2:</strong> At x = e (an integer), we have:</p><p>P(e) - 3 = (e-a)(e-b)(e-c)(e-d)·Q(e)</p><p>Since P(e) = 5:</p><p>5 - 3 = (e-a)(e-b)(e-c)(e-d)·Q(e)</p><p>2 = (e-a)(e-b)(e-c)(e-d)·Q(e)</p><p><strong>Step 3:</strong> The product (e-a)(e-b)(e-c)(e-d) is a product of four distinct non-zero integers. The only ways to factor 2 as a product of integers are: 2 = 2·1 = 1·2 = (-1)·(-2) = (-2)·(-1). Since we need four distinct integer factors whose product (times Q(e)) equals 2, and a, b, c, d are distinct from e, the absolute value of their product must divide 2.</p><p><strong>Step 4:</strong> For four distinct integers with product ±1 or ±2, the minimum configuration is factors like {-1, 1} appearing among the differences. This severely restricts which integers can be a, b, c, d relative to e, making e unique given the roots.</p><p>∴ Answer: D</p>
Correct Answer: D