Lecture Notes: Understanding Polynomials and Like Terms

Introduction to Polynomials

• Polyn means more than one.
• Polynomials consist of multiple terms.
• Discussion on adding and subtracting polynomials, specifically focusing on 'like terms'.

Like Terms

• Definition: Terms that can be combined in addition or subtraction.
• Criteria for Like Terms:
  • Must have the same letters and exponents.
  • Coefficients (numbers in front) can differ.
• Examples:
  • 5x and 3x are like terms.
  • 7y and 4y are like terms.
  • 8xy and 5xy are like terms.
  • 9x² and 2x² are like terms.
  • A standalone number with no variable, like 4, may not have a like term.

Adding and Subtracting Like Terms

• Process:
  • Only the coefficients are added or subtracted.
  • The variables and exponents remain unchanged.
• Examples:
  • 2x + 5x = 7x
  • 3a²b - 7a²b = -4a²b
  • -4x² + 7x² = 3x²
  • 4a - 9a + 6b - 2b = -5a - 4b

Important Note

• Descending Order: Order terms from highest to lowest degree by exponent when writing a polynomial.

Working with Parentheses

• Changing Signs:
  • When a minus sign precedes parentheses, change all signs inside the parentheses.
  • Convert it to a plus sign and change subsequent signs to solve accurately.

Distributing Multiplication

• Distributing: Multiplying each term inside parentheses by the number outside.
• Note: You do not need like terms to multiply.
• Example: Distribute 5 into 2x - 9 = 10x - 45.

Word Problems and Function Rules

• Creating Equations:
  • Translate real-world scenarios into algebraic expressions.
  • View total amounts as functions of variables like months or hours.
• Examples:
  • John’s Savings: T = 36 + 20n (where n is months).
  • Suzanne’s Earnings: A = 100 + 15h (where h is hours babysitting).

Pricing and Cost Functions

• Cable and Gym Charges:
  • Understanding fixed fees vs. monthly charges.
  • Function expressions to calculate total costs over time.
• Examples:
  • Cable: Charges include a setup fee and monthly charges.
  • Gym: Compare different gym pricing structures.

Strategic Difference Calculations

• Handling Differences:
  • Subtract one function from another to find differences in cost structures.
  • Apply the sign change rule when dealing with subtraction in functions.
• Exercise: Practice with Sprint vs. Verizon phone services.

Conclusion

• Key Takeaways:
  • Mastering the concept of like terms is crucial for algebraic operations.
  • Correctly translating word problems into algebraic expressions aids in solving real-world scenarios.
  • Always remember to organize terms in descending order for clarity and correctness.
• Final Thoughts: Paying attention to details like sign changes and term orders can prevent common errors in algebra.