Polynomial Long Division Lecture Notes

Introduction to Polynomial Long Division

• Understanding polynomial long division through a simple numerical example (297 divided by 14).
• Steps of division:
  • Determine how many times the divisor fits into the initial segment of the dividend.
  • Multiply and subtract to find the remainder.
  • Continue the process until all terms are addressed.

Example A: Numerical Division

• 14 goes into 29 twice (2 x 14 = 28).
• Remainder = 29 - 28 = 1.
• Bring down the next digit (7), so now we have 17.
• 14 goes into 17 once (1 x 14 = 14).
• Remainder = 17 - 14 = 3.
• Final result = 21 and 3/14.

Example B: Polynomial Long Division

• Dividing 2x³ by (2x + 5).
• Set up:
  • 2x + 5 outside, 2x³ inside.
• First term: 2x fits into 2x³ how many times? (2x³ / 2x = x²).
• Multiply: x²(2x + 5) = 2x³ + 5x².
• Subtract (distributing the minus): 2x³ - 2x³ = 0, -7x² - 5x² = -12x².
• Bring down the next term (12x).

Continuing Polynomial Division

• -12x² divided by 2x gives -6x.
• Distribute: -6x(2x + 5) = -12x² - 30x.
• Subtract to find remainder: -12x² + 12x² = 0 and add terms: -6x + 30x = 24x.
• Finally, determine how many times 2x goes into 18x: 9.
• No remainder indicates that (2x + 5) is a factor of the cubic polynomial.

Example C: Adjusting for Missing Terms

• When dividing polynomials, ensure all degrees are present (e.g., x², 0, x, constant).
• If terms are missing, use zero placeholders.

Synthetic Division Introduction

• Synthetic division simplifies the process, mainly for linear factors (e.g., x + 5).
• Calculate the zero of the factor: x + 5 = 0 -> x = -5.
• Set up synthetic division using coefficients: 4, 23, -16.

Steps in Synthetic Division

1. Bring down the leading coefficient.
2. Multiply by the zero and add down in each step.
3. Continue until completion.
4. Remainder indicates if the linear factor is a root.

Example D: Synthetic Division with a Cubic Polynomial

• Polynomial: 2x³ - 3x² - 11x + 6 divided by x - 3.
• Use x = 3 for setup.
• Coefficients: 2, -3, -11, 6.
• Process involves bring down, multiply, add down iteratively.

Finalizing the Result

• Coefficients from synthetic division create the quotient polynomial.
• If the remainder is zero, the divisor is a factor of the original polynomial.
• Example: Factor the polynomial into linear terms after synthetic division.

Common Errors to Avoid

• Always ensure all polynomial terms are present; missing terms lead to incorrect results.
• In long division, the structure allows for catching missing terms, while in synthetic division, it's easier to miss them.

Conclusion

• Polynomial long division and synthetic division are essential skills in algebra.
• Practice setting up and executing both methods to become proficient.