
Algebra Brain Summary Malcolm E. Hays 28
October 2002 
This is a list of all the
thoughts located in the Algebra Brain. Each thought is followed by a
statement indicating the content associated by that thought. Thought followed
by (jump) are jump thoughts, which are located on the left hand side of the
central thought in the Brain matrix.
Jump thoughts often take the user to entirely different sections of
the Brain or provide reference information about the central thought. This compilation of
thoughts only includes those thoughts listed under Content in
the Algebra Brain. Algebra Glossary (jump)  Several definitions of common terms used
throughout Algebra Algebra Alphabet – A list of how each letter of
the alphabet is used throughout Algebra Fundamentals of Algebra (jump) – The basics needed to fully understand what Algebra is all about Course Content (jump) – The 5
basic concepts covered in all Algebra courses, from 8th grade to 13th grade Algebraic Expressions – The expression is one of the basic units
used in Algebra, is part of an algebraic function Basic Rules of Algebra – Covers many of the fundamental properties
exhibited by different operators in Algebra Properties of Equality – What equality really means Properties of Exponents – How to manipulate exponents to simplify
algebraic expressions Exponential
Functions (jump) – Basic definition of the exponential
function Properties of Fractions – How to manipulate fractions to simplify
algebraic expressions Rational
Functions (jump) – One polynomial function divided by
another Properties of Inequalities – A list of common properties
that are obeyed by inequalities Linear
Inequalities (jump) – How to solve an inequality
instead of an equation Properties of Zero – Fundamentally, very, very important in
order to fully understand Algebra Cartesian Plane – A brief review of the Cartesian
Plane, including the Distance and Midpoint Formulas Radicals – An introduction to square roots, cube
roots, and other roots Properties of Radicals – How to manipulate radicals to simplify
algebraic expressions Real Numbers – Introduction to different number sets used
throughout Algebra Absolute Value – Introduction to the absolute value
operator and what it really means Ordering Real Numbers – How real numbers relate to each other with respect
to the origin Functions  Introduction
to one of the fundamental concepts in Algebra Function Terminology
(jump) – A
list of common terms associated with functions Inverse Functions  Interchanging
the domain and range of a function Composition of Functions (jump) – A
function of a function Finding the Inverse  How to find the
inverse of a function Horizontal Line Test  The test which
determines if a function has an inverse Polynomial Functions  Functions
involving a sum of powers of x Fundamental Theorem of Algebra (jump)  The number of
roots of a polynomial function equals the highest power of x Complex Numbers  Numbers the complex plane, of which the
real numbers are a subset Polynomial Operations  How to manipulate
polynomials Adding/Subtracting  How to add and
subtract two (or more) polynomials Multiplying
Polynomials  How to
multiply two (or more) polynomials Special
Product Patterns (jump)  A list
of commonly occurring products that occur in Algebra Polynomial
Division  How to
divide two polynomials Synthetic
Division 
Technique used to divide polynomials of a particular form Factor
Theorem  A polynomial f(x) has a factor (x
– k) if and only if f(k) = 0 Remainder
Theorem  If a polynomial f(x) is divided by (x
– k), then the remainder is r = f(k) Real Zeros  The real
solutions of a polynomial equations (as opposed to the complex solutions) Descartes's Rule of Signs  How to determine the number of
positive and negative roots of a polynomial equation Rational
Zero Test  How to
find possible roots of a polynomial function with integer coefficients Special Factoring Patterns  A list of
commonly occurring factoring patterns that occur in Algebra Special Product Patterns  A list of
commonly occurring products that occur in Algebra Multiplying
Polynomials (jump)  How to
multiply two (or more) polynomials Rational Functions  One
polynomial function divided by another Conic Sections (jump) – A
look at some practical examples of rational functions Polynomial Division (jump) – Intimately tied with rational functions Properties of Fractions (jump) – Necessary for understanding how rational
functions work Asymptotes – What they are and how they are used to
help sketch rational functions Partial Fractions  How to convert a
rational function into a sum of two fractions Distinct
Linear Factors 
Denominator of rational function is composed of distinct linear factors Distinct
Quadratic Factors 
Denominator of rational function is composed of irreducible quadratic factors Mixed
Factors – Denominator of rational function contains
both linear and quadratic factors (distinct and/or repeating) Repeated
Linear Factors  Linear
factors that repeat themselves in the denominator of a rational function Repeated
Quadratic Factors  Irreducible
repeating quadratic factors in the denominator of a rational function Sketching Rational Functions  How to sketch the
graph of a rational function Transcendental Functions  A
category of functions that includes both exponential and logarithmic
functions Exponential Functions  Basic definition
of the exponential function Properties
of Exponents (jump) – The
various rules governing the behavior of exponents Exponential
/ Logarithmic Equations – How to use logarithms to solve mathematical
equations Logarithmic/Exponential
Models  Applications of
logarithmic/exponential functions Exponential
Growth and Decay  Very
common application of logs Gaussian  Model which
produces the bellshaped curve used in statistical analysis Logarithmic  Used in a wide
variety of applications including earthquakes, sound, and time of death Logistics
Growth  Model used to accurately represent
population growth in an environment Logarithmic Functions – The inverse of an exponential function Properties
of Logarithms (jump)  Various
rules governing behavior of logarithmic functions Exponential
/ Logarithmic Equations – How to use logarithms to solve mathematical
equations Logarithmic/Exponential
Models  Applications of
logarithmic/exponential functions Exponential
Growth and Decay  Very
common application of logs Gaussian  Model
which produces the bellshaped curve used in statistical analysis Logarithmic  Used in a wide
variety of applications including earthquakes, sound, and time of death Logistics
Growth  Model used to accurately represent
population growth in an environment Natural Base e  Definition of logarithm of base e Translations and Combinations  Moving graphs around the
coordinate plane Arithmetic Combinations  How to add,
subtract, multiply, and divide two functions Composition of Functions  A function of a
function Reflections in Coordinate Axes  How to reflect
a function around either the vertical or horizontal axis Vertical and Horizontal Shifts  How to
move a function vertically and horizontally around a graph Quadratics  Seconddegree
polynomial functions Conic Sections  Classification
of conic sections by the discriminant of a quadratic function Circles  Definition of a circle and its related
equation Ellipses  Definition of ellipse and its related
equation Hyperbolas  Definition of hyperbola and its related
equation Parabolas  Definition of a parabola and its related
equation Translations of Conics  Moving the graph
of a conic section around in the plane Circle
Translation  How to
move a circle around in the plane Ellipse
Translation  How to move
an ellipse around in the plane Hyperbola
Translation  How to
move a hyperbola around in the plane Parabola
Translation  How to
move a parabola around in the plane Quadratic Equations  One of
the most important applications of polynomial functions 1. Factoring  Factor
the quadratic, then set each factor equal to zero to find the roots 2. Extracting Square
Roots  Extracting a square root from a quadratic
equation 3. Completing the Square  Transforming a
quadratic into a perfect square 4. Quadratic Formula  The granddaddy of all methods of solving a quadratic
when nothing else works Linear Equations – Equations that
involve variables raised to the first power only
Linear Inequalities  How to solve
an inequality instead of an equation Absolute Value Inequalities  Inequalities
involving absolute value operator Polynomial Inequalities  How to find test
intervals for solving polynomial inequalities Rational
Inequalities – Solving inequalities that
involve one polynomial divided by another Properties of Inequalities (jump)  A list of common
properties that are obeyed by inequalities Linear Systems  How to
solve more than one linear equation at a time, often involving two or more
variables Linear Systems in a Matrix (jump) – Relationship between linear equations and
matrices Graphical Approach  How to solve a
linear system from a graph of the equations Graphical
Interpretation of Systems (jump) – What linear
systems have to do with the real world Method of Elimination  How to solve a
linear system of equations by eliminating a variable Multivariable Linear Systems  Transformation of
a linear system into rowechelon form Substitution Method  Summary of how to
solve a system of linear equations using backsubstitution Systems of Inequalities – Solving
and graphing systems of linear inequalities Lines and Slope  Various
formulas involving lines and their slopes Equations of Lines  Several different
ways of writing equations of lines Parallel and Perpendicular Lines  Basic
definitions PointSlope Form of a Line  One form of an equation representing a line SlopeIntercept Form of a Line  One form of an
equation representing a line Matrices and Determinants  Basic
definition of a matrix Determinants  A
special operation performed on a matrix Properties of Determinants (jump) – A
list of properties that we can use to simplify the process of finding
determinants Applications of Determinants  Several useful
ways to apply matrices Area of
Triangle  How to
find the area of a triangle with the given vertices using determinants Cramer's
Rule  Solving a system of linear equations using
determinants Lines in Plane  How to
tell if three points lie on the same line and a method for finding the
equation of a line Triangular Matrix  A handy form of
matrix used to quickly solve linear systems of equations Inverse of
a Square Matrix  Analogous to the inverse of a function Finding Matrix Inverse  How to
find the inverse of a matrix Inverse of a 2 x 2 Matrix  Basic
quick formula for finding the inverse of a simple square matrix Linear
Systems in a Matrix – Using a matrix to represent a linear system of equations Elementary Row Operations  One
method of solving systems of linear equations using matrices Linear Systems (jump)  How to solve more than one linear equation at a time Gaussian Elimination – Matrices – Using ERO to find the solution of a system
of linear equations GaussJordan
Elimination – Similar to Gaussian elimination but it
goes a step further Matrix
Operations  List of ways of representing matrices Identity Matrix  A matrix consisting of 1's along its diagonal and zeros
everywhere else Matrix Addition  How
to add two matrices together Matrix Multiplication  How to
multiply two matrices together Properties of Matrix Operations  Properties of matrix addition, matrix
multiplication and scalar multiplication Scalar Multiplication 
Multiplication of a matrix by a constant (scalar) Sequences and Probability  Introductory page about sequences, series,
and probability
Binomial Theorem  Expanding
a binomial raised to an integer power Pascal's Triangle  A more visual way
to find binomial coefficients Counting Principle  Method
of finding total number of ways a particular event can occur Combinations – Number
of ways a group of objects can be arranged irregardless of order Permutations – Number
of ways a group of objects can be arranged, with order being important Factorial – A
special type of function used to multiply integers together in a particular
fashion Probability  Terminology
used throughout probability problems Independent Events  An event that has
no bearing on any previous or subsequent events Union of Two Events  Also defined as
the probability of events A or B occurring Complementary Events  Probability of a
complement is sort of the left over of the probability of an event Sequences – An
arrangement of numbers in a particular order based on a relationship between
those numbers Arithmetic Sequence – A sequence whose terms all have a common
difference between consecutive terms Geometric Sequence – A sequence whose consecutive terms have a
common ratio Summation Notation – A shortcut method of representing the sum
of a large sequence of numbers 
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