<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) when both polynomials leave the same remainder upon division by (x - 4). Set up an equation using P(4) = Q(4) to solve for k.
<p><strong>Step 1:</strong> Apply the Remainder Theorem. When P(x) is divided by (x - 4), the remainder is P(4). When Q(x) is divided by (x - 4), the remainder is Q(4).</p><p><strong>Step 2:</strong> Since both polynomials leave the same remainder: P(4) = Q(4)</p><p><strong>Step 3:</strong> Calculate P(4):</p><p>P(4) = k(4)³ + 3(4)² - 3 = 64k + 48 - 3 = 64k + 45</p><p><strong>Step 4:</strong> Calculate Q(4):</p><p>Q(4) = 2(4)³ - 5(4) + k = 128 - 20 + k = 108 + k</p><p><strong>Step 5:</strong> Set P(4) = Q(4):</p><p>64k + 45 = 108 + k</p><p>63k = 63</p><p>k = 1</p><p>∴ Answer: D (k = 1)</p>
Correct Answer: D