<p>The quadratic polynomial \(p(x)\) has the following properties:</p><ul><li>\(p(x)\) can be positive or zero for all real numbers</li><li>\(p(1) = 0\) and \(p(2) = 2\)</li></ul><p>Then find the quadratic polynomial.</p>
Step-by-Step Solution
Key Concept: Since p(x) ≥ 0 for all real x and p(1) = 0, the polynomial must have a double root at x = 1, making it of the form a(x-1)². Use p(2) = 2 to find the leading coefficient.
<p><strong>Step 1:</strong> Since p(x) ≥ 0 for all real x, the parabola opens upward (a > 0) and its discriminant must be ≤ 0. Since p(1) = 0 and p(x) ≥ 0 for all x, the minimum value occurs at x = 1, meaning x = 1 is a double root.</p><p><strong>Step 2:</strong> Write p(x) = a(x - 1)² where a > 0.</p><p><strong>Step 3:</strong> Use the condition p(2) = 2: a(2 - 1)² = 2, so a(1) = 2, giving a = 2.</p><p><strong>Step 4:</strong> Verify: p(x) = 2(x - 1)² has p(1) = 0 ✓, p(2) = 2(1)² = 2 ✓, and p(x) ≥ 0 for all x ✓</p><p>∴ Answer: p(x) = 2(x - 1)²</p>
Correct Answer: 2(x-1)^2