Lecture Notes: Long Division and Polynomial Division

Introduction to Long Division

• Reminder of the process of long division: dividing a number and understanding the remainder.
• Example revisited: dividing numbers like 22 by 3.
  • Converted into improper fraction or decimal.
  • Relationship between improper fraction and mixed number.

Vocabulary in Division

• Dividend: the number being divided (e.g., 22).
• Divisor: the number you are dividing by (e.g., 3).
• Quotient: the result of division (e.g., 7).
• Remainder: the leftover after division (e.g., 1).

Polynomial Division

• Similar process to numerical long division applied to polynomials.
• Example: dividing polynomial expressions.
  • Expression: (\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)})
  • Remainder typically a constant.

Factoring Polynomials

• Goal: Factor polynomials of degree > 2.
• Process involves breaking down expressions into smaller factors.
• Similar to factoring numbers into prime factors.

Polynomial Long Division Steps

• Similar to numerical division; setup involves dividing the leading term of the dividend by the leading term of the divisor.
• Important to include placeholders for missing powers of x (e.g., 0x).
• Steps: divide, multiply, subtract, bring down the next term, and repeat.

Example: Polynomial Division

• Given: (2x^3 + 3x^2 - 4) divided by (x - 1).
• Process demonstrated with long division setup.
• Resulting quotient and remainder captured.

Synthetic Division

• Efficient method of division using only coefficients.
• Setup with an upside-down division sign.
• Process: bring down, multiply, subtract, repeat.
• Can be used as an alternative to polynomial long division.
• Two methods: keeping the sign or changing it and adding instead of subtracting.

Application of Polynomial Division

• Example problem involving the volume of a rectangular prism.
• Given volume and one dimension, divide to find expressions for the other dimensions.

Remainder Theorem

• States: Remainder when (P(x)) is divided by (x - a) is (P(a)).
• Allows finding remainder without complete division.
• Confirmed with synthetic division.

Summary and Conceptual Understanding

• Polynomial division helps in factoring polynomials and finding roots.
• The lecture covered both theoretical and practical aspects of division.
• Importance of understanding the concepts for solving complex problems.