Polynomial Long Division

Step 1: Set up long division: ________ x + 3 | x² + 7x + 12

Step 2: Divide leading terms: x² ÷ x = x Write x above the division bar

Step 3: Multiply and subtract: x(x + 3) = x² + 3x (x² + 7x + 12) - (x² + 3x) = 4x + 12

Step 4: Bring down and repeat: 4x ÷ x = 4 4(x + 3) = 4x + 12 (4x + 12) - (4x + 12) = 0

Step 5: Write the result: Quotient: x + 4 Remainder: 0

Answer: x + 4

Step 1: Set up long division: ________ x + 3 | x² + 7x + 12

Step 2: Divide leading terms: x² ÷ x = x Write x above the division bar

Step 3: Multiply and subtract: x(x + 3) = x² + 3x (x² + 7x + 12) - (x² + 3x) = 4x + 12

Step 4: Bring down and repeat: 4x ÷ x = 4 4(x + 3) = 4x + 12 (4x + 12) - (4x + 12) = 0

Step 5: Write the result: Quotient: x + 4 Remainder: 0

Answer: x + 4

Step 1: Set up long division: _______________ x + 2 | 2x³ + 5x² - 3x + 1

Step 2: First division: 2x³ ÷ x = 2x² 2x²(x + 2) = 2x³ + 4x² Subtract: (2x³ + 5x²) - (2x³ + 4x²) = x²

Step 3: Second division: x² ÷ x = x x(x + 2) = x² + 2x Subtract: (x² - 3x) - (x² + 2x) = -5x

Step 4: Third division: -5x ÷ x = -5 -5(x + 2) = -5x - 10 Subtract: (-5x + 1) - (-5x - 10) = 11

Step 5: Write result: Quotient: 2x² + x - 5 Remainder: 11

Step 6: Express as: (2x³ + 5x² - 3x + 1) = (x + 2)(2x² + x - 5) + 11

Or: 2x² + x - 5 + 11/(x + 2)

Answer: 2x² + x - 5 with remainder 11

Step 1: Set up long division: _______________ x + 2 | 2x³ + 5x² - 3x + 1

Step 2: First division: 2x³ ÷ x = 2x² 2x²(x + 2) = 2x³ + 4x² Subtract: (2x³ + 5x²) - (2x³ + 4x²) = x²

Step 3: Second division: x² ÷ x = x x(x + 2) = x² + 2x Subtract: (x² - 3x) - (x² + 2x) = -5x

Step 4: Third division: -5x ÷ x = -5 -5(x + 2) = -5x - 10 Subtract: (-5x + 1) - (-5x - 10) = 11

Step 5: Write result: Quotient: 2x² + x - 5 Remainder: 11

Step 6: Express as: (2x³ + 5x² - 3x + 1) = (x + 2)(2x² + x - 5) + 11

Or: 2x² + x - 5 + 11/(x + 2)

Answer: 2x² + x - 5 with remainder 11

Step 1: Rewrite dividend with all terms: x³ + 0x² + 0x - 27

Step 2: Set up and perform long division: x² + 3x + 9 _______________ x - 3 | x³ + 0x² + 0x - 27

Step 3: First division: x³ ÷ x = x² x²(x - 3) = x³ - 3x² Subtract: (x³ + 0x²) - (x³ - 3x²) = 3x²

Step 4: Second division: 3x² ÷ x = 3x 3x(x - 3) = 3x² - 9x Subtract: (3x² + 0x) - (3x² - 9x) = 9x

Step 5: Third division: 9x ÷ x = 9 9(x - 3) = 9x - 27 Subtract: (9x - 27) - (9x - 27) = 0

Step 6: Recognize the pattern: This is the difference of cubes: a³ - b³ = (a - b)(a² + ab + b²) x³ - 27 = (x - 3)(x² + 3x + 9)

Answer: x² + 3x + 9

Step 1: Rewrite dividend with all terms: x³ + 0x² + 0x - 27

Step 2: Set up and perform long division: x² + 3x + 9 _______________ x - 3 | x³ + 0x² + 0x - 27

Step 3: First division: x³ ÷ x = x² x²(x - 3) = x³ - 3x² Subtract: (x³ + 0x²) - (x³ - 3x²) = 3x²

Step 4: Second division: 3x² ÷ x = 3x 3x(x - 3) = 3x² - 9x Subtract: (3x² + 0x) - (3x² - 9x) = 9x

Step 5: Third division: 9x ÷ x = 9 9(x - 3) = 9x - 27 Subtract: (9x - 27) - (9x - 27) = 0

Step 6: Recognize the pattern: This is the difference of cubes: a³ - b³ = (a - b)(a² + ab + b²) x³ - 27 = (x - 3)(x² + 3x + 9)

Answer: x² + 3x + 9

Step 1: Note missing x⁴ and x terms in dividend: 4x⁴ + 0x³ - 8x² + 0x + 5

Step 2: Set up division (dividing by 2x² + 1): 2x² - 5 ________________ 2x² + 1 | 4x⁴ + 0x³ - 8x² + 0x + 5

Step 3: First division: 4x⁴ ÷ 2x² = 2x² 2x²(2x² + 1) = 4x⁴ + 2x² Subtract: (4x⁴ - 8x²) - (4x⁴ + 2x²) = -10x²

Step 4: Second division: -10x² ÷ 2x² = -5 -5(2x² + 1) = -10x² - 5 Subtract: (-10x² + 5) - (-10x² - 5) = 10

Step 5: Write result: Quotient: 2x² - 5 Remainder: 10

Answer: 2x² - 5 + 10/(2x² + 1)

Step 1: Note missing x⁴ and x terms in dividend: 4x⁴ + 0x³ - 8x² + 0x + 5

Step 2: Set up division (dividing by 2x² + 1): 2x² - 5 ________________ 2x² + 1 | 4x⁴ + 0x³ - 8x² + 0x + 5

Step 3: First division: 4x⁴ ÷ 2x² = 2x² 2x²(2x² + 1) = 4x⁴ + 2x² Subtract: (4x⁴ - 8x²) - (4x⁴ + 2x²) = -10x²

Step 4: Second division: -10x² ÷ 2x² = -5 -5(2x² + 1) = -10x² - 5 Subtract: (-10x² + 5) - (-10x² - 5) = 10

Step 5: Write result: Quotient: 2x² - 5 Remainder: 10

Answer: 2x² - 5 + 10/(2x² + 1)

Step 1: Use the Remainder Theorem: When P(x) is divided by (x - a), remainder = P(a) Here, a = 1 and remainder = 8

Step 2: Set up the equation: P(1) = 8 3(1)³ + k(1)² - 5(1) + 2 = 8

Step 3: Simplify: 3 + k - 5 + 2 = 8 k + 0 = 8 k = 8

Step 4: Verify by long division (optional): With k = 8: P(x) = 3x³ + 8x² - 5x + 2 Divide by (x - 1):

       3x² + 11x + 6
  ___________________

x - 1 | 3x³ + 8x² - 5x + 2

First: 3x³ ÷ x = 3x², then 3x²(x-1) = 3x³ - 3x² Subtract: 8x² - (-3x²) = 11x²

Second: 11x² ÷ x = 11x, then 11x(x-1) = 11x² - 11x Subtract: -5x - (-11x) = 6x

Third: 6x ÷ x = 6, then 6(x-1) = 6x - 6 Subtract: 2 - (-6) = 8 ✓

Answer: k = 8

Step 1: Use the Remainder Theorem: When P(x) is divided by (x - a), remainder = P(a) Here, a = 1 and remainder = 8

Step 2: Set up the equation: P(1) = 8 3(1)³ + k(1)² - 5(1) + 2 = 8

Step 3: Simplify: 3 + k - 5 + 2 = 8 k + 0 = 8 k = 8

Step 4: Verify by long division (optional): With k = 8: P(x) = 3x³ + 8x² - 5x + 2 Divide by (x - 1):

       3x² + 11x + 6
  ___________________

x - 1 | 3x³ + 8x² - 5x + 2

First: 3x³ ÷ x = 3x², then 3x²(x-1) = 3x³ - 3x² Subtract: 8x² - (-3x²) = 11x²

Second: 11x² ÷ x = 11x, then 11x(x-1) = 11x² - 11x Subtract: -5x - (-11x) = 6x

Third: 6x ÷ x = 6, then 6(x-1) = 6x - 6 Subtract: 2 - (-6) = 8 ✓

Answer: k = 8