Math 100: Polynomials in Several Variables

Introduction

• Polynomials involve more than one variable (e.g., X and Y).
• Example terms:
  • 3x³Y: 3 times X cubed times Y
  • XY²: X times Y squared
  • 5Y: 5 times Y

Evaluating Polynomials

• Evaluate: Substitute values to get a numerical answer.
• Example:
  • Substitute X = -1, Y = 5 into polynomial expressions.
  • Follow order of operations (PEMDAS):
    1. Exponents
    2. Multiplication
    3. Addition/Subtraction left to right
• Calculate step-by-step to solve the expression.
  • Final result in example: -9

Coefficient and Degree

• Coefficient: The numerical factor in a term.
• Degree of Term: Sum of the exponents in the term.
• Degree of Polynomial: Highest degree among the terms.
• Example Polynomial: 8x⁴y⁵ - 7x³y² - 5x + 11
  • Create a table with terms, coefficients, and degrees.
  • Identify the greatest degree in the terms for the polynomial's degree.

Adding and Subtracting Polynomials

• Identify like terms:
  • Like terms have identical variable parts.
• Example addition:
  • Combine like terms from 6xy² - 5xy + 7 and 9xy² + 2xy - 6
  • Result: 15xy² - 3xy + 1
• Example subtraction:
  • Work with subtraction signs carefully, especially with "minus a negative."
  • Combine like terms appropriately.
  • Result: 3x³ + 2x²y + 5xy² - 10

Multiplying Polynomials

• Use the Distributive Property:
  • Multiply each term in the first polynomial by every term in the second.
• Example Multiplication:
  • Multiply (x - y) * (x² + xy + y²)
  • Distribute and combine like terms.
  • Result: x³ - y³

Conclusion

• Stay on pace with the pacing guide.
• Email questions if any doubts arise.