<p>If \((x+1)\) is a factor of \(x^3 + kx^2 - 3x + k + 2\), then \(k\) is equal to:</p>
Step-by-Step Solution
Key Concept: If (x+1) is a factor, then the polynomial equals zero when x = -1 (Factor Theorem). Substitute x = -1 and solve for k.
<p><strong>Step 1:</strong> Apply Factor Theorem. If (x+1) is a factor, then P(-1) = 0.</p><p><strong>Step 2:</strong> Substitute x = -1 into P(x) = x³ + kx² - 3x + k + 2:<br/>P(-1) = (-1)³ + k(-1)² - 3(-1) + k + 2 = 0</p><p><strong>Step 3:</strong> Simplify:<br/>-1 + k + 3 + k + 2 = 0<br/>2k + 4 = 0<br/>2k = -4<br/>k = -2</p><p><strong>Step 4:</strong> Verify: P(x) = x³ - 2x² - 3x becomes x³ - 2x² - 3x, and P(-1) = -1 - 2 + 3 = 0 ✓</p><p>∴ Answer: D (k = -2)</p>
Correct Answer: D