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Understanding Polynomial Long Division Techniques

Mar 10, 2025

Polynomial Long Division Overview

Basic Long Division Review

  • Example: 297 divided by 14
    • 14 goes into 29 (2 times)
    • 2 x 14 = 28
    • Subtract: 29 - 28 = 1
    • Bring down 7, making it 17
    • 14 goes into 17 (1 time)
    • Subtract: 17 - 14 = 3
    • Result: 21 and 3/14

Long Division with Polynomials

Example 1: 2x³ ÷ (2x + 5)

  1. Setup: 2x + 5 goes outside, 2x³ inside.
  2. First term division:
    • 2x into 2x³ = x²
    • Multiply: (2x + 5) x x² = 2x³ + 5x²
  3. Subtract:
    • Result: 2x³ - 2x³ = 0
    • -7x² - 5x² = -12x²
  4. Bring down 12x.
  5. Repeat process:
    • 2x into -12x² = -6x
    • Multiply: -6x(2x + 5) = -12x² - 30x
  6. Subtract:
    • Result: -12x² + 12x² = 0
    • 12x + 30x = 42x
    • Bring down remaining terms.
  7. Final division:
    • 2x into 18x = 9
    • Remainder: 0 (thus, 2x + 5 is a factor)

Example 2: Missing Terms

  • Ensure all polynomial degrees are represented (insert zeros for missing terms).
  • Example: x² terms missing in the polynomial.

Writing the Remainder

  • When not a factor:
    • Remainder format: (3x² - 4x + 9) + (4x - 4)/(x² + 1)

Synthetic Division

  • Use for linear factors.
  1. Set x + 5 to zero: x = -5.
  2. Coefficients: (4, 23, -16) for the polynomial.
  3. Process:
    • Bring down first coefficient.
    • Multiply and add down the line.
  4. Final result: Quotient and remainder.

Example 3: Synthetic Division with a Cubic Polynomial

  1. Divide 2x³ - 3x² - 11x + 6 by x - 3.
  2. Setup with coefficients.
  3. Bring down and use addition for each step.
  4. Remainder of 0 indicates that x - 3 is a factor.
  5. Coefficients obtained lead to the quotient polynomial (2x² + 3x - 2).
  6. Factor further if possible.

Common Pitfalls

  • Missing terms in polynomial division can lead to incorrect answers.
  • Always check for zero placeholders in synthetic division setup.

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