Polynomials and Factorisation

Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: Look at each term: 3x⁴ (degree 4), -5x² (degree 2), 7x (degree 1), -2 (degree 0) Step 3: The highest degree is 4. Answer: Degree = 4

Answer

Step 1: Substitute x = 2 in the polynomial p(x) = 2x³ - 3x + 1 Step 2: p(2) = 2(2)³ - 3(2) + 1 Step 3: p(2) = 2(8) - 6 + 1 Step 4: p(2) = 16 - 6 + 1 = 11 Answer: p(2) = 11

Answer

Step 1: Set the polynomial equal to zero: 3x - 15 = 0 Step 2: Add 15 to both sides: 3x = 15 Step 3: Divide both sides by 3: x = 5 Step 4: Verify: 3(5) - 15 = 15 - 15 = 0 ✓ Answer: x = 5

Answer

Step 1: By remainder theorem, remainder = p(a) where divisor is (x - a) Step 2: Here divisor is (x - 1), so a = 1 Step 3: p(x) = x³ + 2x² - x + 3 Step 4: p(1) = (1)³ + 2(1)² - (1) + 3 Step 5: p(1) = 1

Answer

Step 1: Use factor theorem - if (x + 2) is a factor, then p(-2) = 0 Step 2: p(x) = x³ + 3x² + 3x + 2 Step 3: p(-2) = (-2)³ + 3(-2)² + 3(-2) + 2 Step 4: p(-2) = -8 + 3(4) - 6 + 2 Step 5: p(-2) = -8 + 1

Answer

Step 1: Find two numbers that multiply to 12 and add to 7 Step 2: Consider pairs: (1,12), (2,6), (3,4) Step 3: Check: 3 + 4 = 7 and 3 × 4 = 12 ✓ Step 4: x² + 7x + 12 = x² + 3x + 4x + 12 Step 5: = x(x

Answer

Step 1: Use identity (a + b)² = a² + 2ab + b² Step 2: Here a = 2x, b = 3 Step 3: (2x + 3)² = (2x)² + 2(2x)(3) + (3)² Step 4: = 4x² + 12x + 9 Answer: 4x² + 12x + 9

Answer

Step 1: Recognize as difference of squares: a² - b² Step 2: 9x² = (3x)² and 25 = (5)² Step 3: Use identity a² - b² = (a + b)(a - b) Step 4: 9x² - 25 = (3x)² - (5)² = (3x + 5)(3x - 5) Answer: (3x + 5)(