Algebra — Important Questions

6

Step 1: By Factor Theorem, if (x - 2) is a factor of p(x), then p(2) = 0. Step 2: Substitute x = 2 in p(x): p(2) = (2)² - 5(2) + k = 0. Step 3: = 4 - 10 + k = 0. Step 4: -6 + k = 0, so k = 6. Step 5: Verify: p(2) = 4 - 10 + 6 = 0. ✓ The value of k is 6.

9x² + 24xy + 16y²

Step 1: Use identity (a + b)² = a² + 2ab + b² with a = 3x and b = 4y. Step 2: a² = (3x)² = 9x². Step 3: 2ab = 2 × 3x × 4y = 24xy. Step 4: b² = (4y)² = 16y². Step 5: So (3x + 4y)² = 9x² + 24xy + 16y². Common mistake: forgetting the middle term 2ab or computing 2×3×4 = 12 instead of 24.

10404

Step 1: Write 102 = 100 + 2. Use (a + b)² = a² + 2ab + b² with a = 100, b = 2. Step 2: a² = 100² = 10000. Step 3: 2ab = 2 × 100 × 2 = 400. Step 4: b² = 2² = 4. Step 5: (102)² = 10000 + 400 + 4 = 10404.