Chapter 3 Notes

Overview of Topics Covered

  • Multiplication in Modeling Problems

    • Discussed in Section 3.2.
    • Different methodologies exist for solving multiplication within modeling contexts.

  • Identities and Factoring

    • Section 3.3 focused on a variety of factoring techniques including:
    • Factoring a Difference of Cubes
      • Formula: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
    • Factoring a Sum of Cubes
      • Formula: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
    • Factoring a Difference of Squares
      • Formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
    • Factoring a Sum of Squares
      • Note: Sum of squares is typically not factorable over the reals.

  • Polynomial Expansion

    • Discussed the Binomial Theorem
    • Connection made to Pascal's Triangle, which provides coefficients for binomial expansions.

  • Assessment Information

    • Tests will include a mix of:
    • Free Response Questions
    • Multiple Choice Questions

Division Techniques in Polynomials

  • Dividing Polynomials

    • Section 3.4 introduced techniques such as:
    • Long Division
      • Key Reminder: Subtraction is used in long division.
    • Synthetic Division
      • Key Reminder: Addition is used in synthetic division.

  • Remainder Theorem

    • Connection made to synthetic division.
    • States that if a polynomial f(x)f(x) is divided by x−cx - c, the remainder is f(c)f(c).

  • Factor Theorem

    • An extension of the Remainder Theorem, it states that x−cx - c is a factor of f(x)f(x) if and only if f(c)=0f(c) = 0.

Exponents and Their Effects on Graphs

  • Exponents and Roots

    • Example: For the polynomial x−4x - 4 raised to the power of 4,
    • Setting it equal to zero: x - 4 = 0
      ightarrow x = 4
    • Interpretation of the Exponent:
      • Even exponent (4) indicates a Turning Point rather than crossing the x-axis.
      • Evens lead to turning points.
      • Odds lead to crossing the axis.

  • Determining Even or Odd Functions

    • Methodology: Substitute with negative xx:
    • If f(−x)=f(x)f(-x) = f(x), then the function is Even.
    • If f(−x)=−f(x)f(-x) = -f(x), then the function is Odd.
    • If only some signs change, the function is Neither.
    • Graph Symmetries:
    • Symmetric about the y-axis: Even
    • Symmetric about the origin: Odd

Graph Characteristics and Behavior

  • End Behavior of Polynomials
    • Example: A function with a constant term of -5 implies
    • The graph shifts down by 5 units at the end.
    • Example: If the function has a term −x-x, this often indicates a leftward movement in the graph.

Review Resources and Test Preparation

  • Review Materials

    • Instructor has created a review resource in Math Excel, which is optional and not collected for a grade.
    • This resource encompasses bulk of potential test questions.

  • Preparation Protocol

    • Students encouraged to utilize past quizzes and Math Excel resources for revision.
    • Questions for clarification invited from students regarding the material covered.

  • Integration of Content

    • Emphasis on the alignment of test questions derived from class notes, Math Excels, and quizzes, ensuring familiarity for students.
    • Students reminded to access Math Excel via the book for practice and review rather than through integrated accounts.

Conclusion

  • Instructor concludes with an invitation for questions and emphasizes the importance of revision in preparation for the upcoming test, ensuring that students are equipped with the necessary tools for success.