Algebra II For Dummies

  • Algebra II For Dummies by Mary Jane Sterling, published by Wiley Publishing, Inc.

  • Copyright © 2006 by Wiley Publishing, Inc.

  • About the Author: Mary Jane Sterling has authored multiple algebra and trigonometry books and has taught math at Bradley University for over 25 years.

  • Dedication: The author dedicates the book to her husband, brothers, and brother-in-law.

  • Acknowledgments: The author thanks Mike Baker, Josh Dials, Alexsis Venter, and Kathy Cox for their contributions to the book.

  • Contents Overview:

    • Part I: Basic Solutions (Chapters 1-5)
    • Part II: Functions (Chapters 6-10)
    • Part III: Conics and Systems of Equations (Chapters 11-13)
    • Part IV: Advanced Concepts (Chapters 14-17)
    • Part V: Part of Tens (Chapters 18-19)

  • Introduction:

    • The book aims to help readers cope with different algebra teaching methods.
    • It covers uncovering mysteries, historical perspectives, providing information, and introducing Algebra II with humor.

  • About This Book:

    • Suitable for those fresh off Algebra I, those returning to algebra, parents of Algebra II students, and those curious about science and mathematics.

  • Conventions Used:

    • Italicized special mathematical terms with definitions.
    • Boldface text for keywords or action parts of numbered steps.
    • Sidebars for interesting but not critical information.

  • Foolish Assumptions:

    • Assumes a grasp of arithmetic of signed numbers, order of operations, solving equations, basic graphs, and basic Algebra I terms.

  • How This Book Is Organized:

    • Part I: Homing in on Basic Solutions. Covers basics, solving equations/factoring, linear equations, quadratics, rational/radical equations, and graphing.
    • Chapters Include: solving equations, inequalities, properties.
    • Part II: Facing Off with Functions. Covers algebraic, exponential, and logarithmic functions, function properties.
    • Chapters Include: properties, domain, rational functions, asymptotes, exponential functions.
    • Part III: Conquering Conics and Systems of Equations. Focuses on graphing and systems.
    • Chapters Include: linear equations, system equations.
    • Part IV: Shifting into High Gear with Advanced Concepts. Includes matrices, sequences, series, and sets.
    • Chapters Include: matrices, sets, sequences.
    • Part V: The Part of Tens. Lists of multiplication tricks and special types of numbers.

  • Icons Used:

    • Rules of the road.
    • Information to improve mind and skills.
    • Points to soak up before proceeding.
    • Alerts to common potential problems.
    • Technical items of interest.

  • Chapter 1 Key Concepts

    • Commutative Property:
    • Addition: a+b=b+aa + b = b + a
    • Multiplication: a⋅b=b⋅aa ⋅ b = b ⋅ a
    • Associative Property:
    • Addition: a+(b+c)=(a+b)+ca + (b + c) = (a + b) + c
    • Multiplication: a(b⋅c)=(a⋅b)ca(b ⋅ c) = (a ⋅ b)c
    • Distributive Property:
    • Multiplication over Addition: a(b+c)=a⋅b+a⋅ca(b + c) = a ⋅ b + a ⋅ c
    • Multiplication over Subtraction: a(b–c)=a⋅b–a⋅ca(b – c) = a ⋅ b – a ⋅ c
    • Additive Identity: a+0=0+a=aa + 0 = 0 + a = a
    • Multiplicative Identity: a⋅1=1⋅a=aa ⋅ 1 = 1 ⋅ a = a
    • Multiplication Property of Zero:
    • If a⋅b⋅c⋅d⋅e⋅f=0a ⋅ b ⋅ c ⋅ d ⋅ e ⋅ f = 0, at least one factor is 0.

  • Exponential Rules:

    • Multiplying with the Same Base: an⋅am=am+na^n ⋅ a^m = a^{m+n}
    • Dividing with the Same Base: anam=an−m\frac{a^n}{a^m} = a^{n-m}
    • Root of Exponents: xn=x1n\sqrt[n]{x} = x^{\frac{1}{n}}
    • Power to a Power: (am)n=am⋅n(a^m)^n = a^{m ⋅ n}
    • Negative Exponents: a−n=1ana^{-n} = \frac{1}{a^n}

  • Factoring Techniques:

    • Greatest Common Factor: ax+ay=a(x+y)ax + ay = a(x + y)
    • Difference of Squares: x2–a2=(x–a)(x+a)x^2 – a^2 = (x – a)(x + a)
    • Difference of Cubes: x3–a3=(x–a)(x2+ax+a2)x^3 – a^3 = (x – a)(x^2 + ax + a^2)
    • Sum of Cubes: x3+a3=(x+a)(x2–ax+a2)x^3 + a^3 = (x + a)(x^2 – ax + a^2)
    • Trinomials: x2n+(a+b)xn+ab=(xn+a)(xn+b)x^{2n} + (a+b)x^n + ab = (x^n + a)(x^n + b)
      Factoring by Grouping
    • Always look for a greatest common factor first
    • Use pairs of terms to group factors to yield a new common factor

  • Chapter 2 Key Concepts

    • Basic linear equation, ax+b=c
    • isolate the variable on one side of the equation
    • fraction removal by multiplying both sides of equasion by LCD
    • Isolating Unknowns
    • Linear Inequalities, such as
    • Absolute Values: |a|
    • Solving absolute-value inequalities.

  • Chapter 3 Key Concepts

    • Simple Quadratics: x2=k,thenx=±kx^2 = k, then x = \pm \sqrt{k}
    • Quadratic Formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
      This works great when factorization isn't immediately obvious or not possible.

  • Completing the Square:

  • A technique to solve quadratic equations and prepare equations for conic sections.

  • Factoring Techniques

    • Grouping, Difference, or Sums of Cubes or Squares methods

  • Solving Quadratic Inequalities

    • Make sure the polynomial is =0 first.
    • Determine zero's (roots).
    • Put zero's in order on # line.
    • Use a sign-line to determine the positive and negative intervals.
    • Determine the overall solution; usually w/ inequality notation.