<p><strong>177.</strong> The polynomials \(P(x) = kx^3 + 3x^2 - 3\) and \(Q(x) = 2x^3 - 5x + k\), when divided by \((x-4)\) leave the same remainder, then \(k\) is equal to:</p>
Step-by-Step Solution
Key Concept: By the Remainder Theorem, P(4) = Q(4) since both polynomials leave the same remainder when divided by (x-4). Set up an equation using this condition to solve for k.
<p><strong>Step 1:</strong> Apply the Remainder Theorem. When P(x) is divided by (x-4), remainder = P(4). When Q(x) is divided by (x-4), remainder = Q(4).</p><p><strong>Step 2:</strong> Since both remainders are equal: P(4) = Q(4)</p><p><strong>Step 3:</strong> Calculate P(4):<br/>P(4) = k(4)³ + 3(4)² - 3<br/>P(4) = k(64) + 3(16) - 3<br/>P(4) = 64k + 48 - 3 = 64k + 45</p><p><strong>Step 4:</strong> Calculate Q(4):<br/>Q(4) = 2(4)³ - 5(4) + k<br/>Q(4) = 2(64) - 20 + k<br/>Q(4) = 128 - 20 + k = 108 + k</p><p><strong>Step 5:</strong> Set P(4) = Q(4):<br/>64k + 45 = 108 + k<br/>64k - k = 108 - 45<br/>63k = 63<br/>k = 1</p><p>∴ Answer: B (k = 1)</p>
Correct Answer: B