<p>Let \(g(x) = (x+2)(x-1)\). If \(f(x) = Q(x)\cdot(x+2)(x-1)+(4x+3)\) and \(f(1) = a+b+6 = 7\), \(f(-2) = 4a-2b+3 = -5\), then which of the following are correct?</p>
Step-by-Step Solution
Key Concept: When a polynomial f(x) is divided by g(x) = (x+2)(x-1), the remainder is a linear expression (4x+3), and we can use the remainder theorem at the roots of g(x) to find unknown coefficients in f(x).
<p><strong>Step 1: Apply Remainder Theorem</strong></p><p>Since f(x) = Q(x)·(x+2)(x-1) + (4x+3), at the roots of g(x):</p><p>f(1) = Q(1)·0 + (4·1+3) = 7 ✓</p><p>f(-2) = Q(-2)·0 + (4·(-2)+3) = -5 ✓</p><p>This confirms the remainder is correct.</p><p><strong>Step 2: Solve the System from Given Conditions</strong></p><p>From f(1) = a+b+6 = 7:</p><p>a + b = 1 ... (equation 1)</p><p>From f(-2) = 4a-2b+3 = -5:</p><p>4a - 2b = -8</p><p>2a - b = -4 ... (equation 2)</p><p><strong>Step 3: Solve Simultaneously</strong></p><p>Adding equations 1 and 2:</p><p>(a+b) + (2a-b) = 1 + (-4)</p><p>3a = -3</p><p>a = -1</p><p>From equation 1: -1 + b = 1, so b = 2</p><p><strong>Step 4: Verify</strong></p><p>f(1) = -1 + 2 + 6 = 7 ✓</p><p>f(-2) = 4(-1) - 2(2) + 3 = -4 - 4 + 3 = -5 ✓</p><p>∴ a = -1, b = 2</p>
Correct Answer: A, B, C, D