Lecture Notes: Operations with Polynomials

Introduction

• Presenter: Robert from YayMath.org
• Topic: Operations with Polynomials
• Aim: To make the concept easy to understand for everyone

Key Concepts

Degree of a Polynomial

• Definition: The highest value of the exponent in a polynomial.
• Examples:
  • ( x^4 ) has degree 4.
  • In subtraction without multiplication, the highest degree is the degree of the polynomial.
  • When multiplying, add the exponents (e.g., ( x^3 \cdot x^4 = x^7 ), degree 7).

Adding and Subtracting Polynomials

• Combine like terms:
  • Distribute the negative when subtracting: Change the sign of each term.
  • Example: ((3x^2 + 2x - 7) - (x^2 - 2x + 10))
    • Change signs: (-x^2 + 2x - 10)
    • Result: (2x^2 + 4x - 17)

Multiplying Polynomials

• Common Mistake: Incorrectly applying the power to each term individually (e.g., ((2x - y)^3) is NOT ((2x)^3 - y^3)).
• Correct Method: Distribute each term separately:

FOIL Method

• First, Outer, Inner, Last
  • Multiply binomials: ((2x - y)(2x - y))
  • Example:
    • First: (2x \cdot 2x = 4x^2)
    • Outer: (2x \cdot (-y) = -2xy)
    • Inner: (-y \cdot 2x = -2xy)
    • Last: (-y \cdot -y = y^2)
  • Combine: (4x^2 - 4xy + y^2)

Extending to Higher Powers

• Multiply the result with the remaining binomial.
• Practical Tip: Write larger to avoid mistakes.
• Example:
  • Result from above: Multiply with another ((2x - y))
  • Combine using like terms to find final expression.

Conclusion

• Multiplying polynomials involves careful step-by-step distribution.
• Avoid shortcuts that lead to mistakes.
• Embrace making mistakes as a learning tool.
• Recommended to write large and use resources to avoid errors.

Note: Use paper efficiently for educational purposes.

Thank you for attending this session on operations with polynomials! Keep practicing for mastery.