• Advanced Algebra Notes
    Updated Notes can be found on the Advanced Algebra Notes link:
    Conic Sections
    Chapter 8 Rationals
    p 533   

    Rational Expressions
    Find common denominators, combine like terms, factor is possible and reduce.
    Multiplying and Dividing  
    Adding and Subtracting Compound Rationals (nested fractions) and negative exponents:
    Adding and Subtracting  

    Finding asymptotes
    Boromir says
    holes and vertical asymptotes  
    Direct Variation
    Joint Combined Variation  

    Rational Zeros/Roots
    Rational Zeros Theorem  
    Chapter 7 Polynomials
    factor with variable sub
    Long Div/finding zeros
    finding poly EQ  
    Synthetically Complex  

    Synthetic Division  

    Polynomial Factor Theorem
    Polynomial Remainders  
    Sum & Difference of Cubes  
    Polynomial Graphing
    Polynomial Graphs  
    Polynomial Intro
    Combining Polynomials  
    Graphing Polynomials in Factored Form: (Polynomial Graphs KEY)
    Polynomial Graphs KEY  
    Changes to FER#2: 
    Change the right hand side of the equation to
    and it should work

    Problem XVII should have a starting amount of 100 bacteria instead of just 1.                                                           

    Chapter 6 Logarithms
     Bank of e
    Lender's National  

    Chapter 3 Systems
    October 15, 2013
    Linear Inequalities Quiz Review and More Mixture Problems
    October 14, 2013
    Linear Constraints 1of2
    Linear Constraints 2of2  
    Chapter 2 Functions
     Chapter 2 Review answers:
     Transformations answer key:
    Do Now
    Piecewise Equations
    Piecewise Graphs  
    Inverses of functions
    fog = x
    proving inverses  
    function notation
    operations of functions
    foggy goffing

    Chapter 1 Foundations Review
    September 9, 2013
    Nine zulu queens rule China but the king awakes and surveys his 
    Domain (horizon from left to right, x-values) and imagines the
    Range (vertical up and down y-values) of possibilities.  "Sky's the limit."  y goes with sky.
    Domain Range notes  
    Exponents are defined as repeated multiplication.
    5^2 means multiply 5 by itself twice, 5*5 = 25.
    2^3 means multiply 2 by itself three times, 2*2*2 = 8.
    This also means add exponents when multiplying identical bases:
    x^2 * x^3 = x*x*x*x*x = x^5 = x^(2+3).
    Likewise, subtract exponents when dividing identical bases:
    a^4 / a^3 = a*a*a*a / (a*a*a) = a (a*a*a) / (a*a*a) = a ^ 1 = a ^ (4-3) = a.
     Properties of Exponents
    Since a negative is the opposite of a positive, a negative exponent means
    do the opposite of multiplying.  So divide.  For example,
    5^(-1) means do the opposite of multiplying by 5 once.  So instead of 1*5,
    do 1/5.  So 5^(-1) = 0.2.
    5^(-2) means divide by 5 twice.  1/5/5 = 1/5^2 = 1/25.
    Rules of Powers
    What about fractional exponents?  
    If 2*2*2 = 8, then
    8^(1/3) = 2 (one third of the product) and
    8^(2/3) = 2*2 = 4 (two thirds of the product.)
    Therefore, 8^(1/3) is the cube root of 8, or 2.
    Square that to get 8^(2/3), or 2^2 = 4.
    Fractional Powers  
    Unit 2
    Notes 2-1.1
    Notes 2-1.2
    Nine Zulu Queens Rule China:
    N = Natural "counting" numbers
    Z = Integers, 0 with positive and negative natural numbers
    Q = Rational numbers, ratios of two integers, Q meaning quotient. 
            (Proper and improper fractions, mixed numbers, repeating and terminating decimals.)
    R = Real numbers, which is the set of Q along with irrational numbers
          (non-repeating non-terminating decimals, e.g. Φ = 1.618033..., √2, 0.12345678910111213..., 0.101001000100001...)
    Notes 2-1.3
    Closure is the property by which the resulting answer is a member of the same set as the two operands.
    For example, adding or multiplying two natural numbers results in another natural number. 
    So the Natural number set is closed under addition and multiplication.  But not subtraction because 2-5 = -3.
    However, the set Z of integers is closed under addition, multiplication, AND subtraction as sums, products, and differences of integers are still integers.  But not quotients.  Sometimes dividing two integers yields a fraction or a decimal, in other words, a rational number.
    Therefore, the set of rationals Q is closed under all four operations (+, -, *, /) and integral powers.  However, inverting powers, i.e. taking roots, often results in an irrational number.  √2, √3, √5, etc. all result in non-terminating non-repeating decimals.  This development was once so controversial it cost a mathematician his life as Hipparchus showed his Pythagoreans colleagues that √2 was not a fraction, and therefore all was not number as they had previously believed every number could be written as a fraction.  They were none too happy with him about this discovery.
    Quite often, a new number set was discovered and developed by inverting an operation on a simpler set
    (subtracting Naturals, dividing Integers, rooting Rationals.)  Indeed, square rooting a negative number is not closed in the set of Reals, which leads to another larger number set denoted by C.  We will learn these imaginary complex numbers in a later chapter.
    Review Unit 1
    Unit 1 Lesson 1 Notesheet 1.1 (Slope Triangles and x- and y-intercepts)
    Slopes & intercepts
     Notesheet 1.2 (Linear Equations and Solving Systems of Equations)
    Linear Equations & Systems
    Unit 1 Lesson 8/22/2013 (Factoring Trinomials, Parallel Lines)
    Factoring & Parallels
    Perpendicular Lines, Graphing Parabolas
    Perpendicular & Parabolas  
    Solving absolute value EQuations and Quadratic EQuations
    Solving abs val & quadratics
Last Modified on September 11, 2017