CLEP College Algebra – Algebra Principles

Algebra
one of the main branches of mathematics, it concerns the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also covers symbols, variables, and set elements. Addition and multiplication are general operations, but their precise definitions lead to structures such as groups, rings, and fields.
algebraic equation
is an equation involving only algebraic expressions in the unknowns. These are further classified by degree.
A linear equation
is algebraic equation of degree one
A polynomial equation
is an equation in which a polynomial is set equal to another polynomial.
A transcendental equation
is an equation involving a transcendental function of one of its variables.
A functional equation
is an equation in which the unknowns are functions rather than simple quantities.
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A differential equation
is an equation involving derivatives.
A integral equation
is an equation involving integrals.
A Diophantine equation
is an equation where the unknowns are required to be integers.
Polynomials
Variables
Algebra
(from Arabic al-jebr meaning “reunion of broken parts”) is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.
Pure mathematics
Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of
Elementary algebra
introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. in which the properties of operations on the real number system are recorded using symbols as “place holders” to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied. This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). University-level courses in group theory may also be called elementary algebra.
categories of Algebra
Elementary algebra
Abstract algebra
Linear algebra
Universal algebra
Algebraic number theory
Algebraic geometry
Algebraic combinatorics
Abstract algebra
sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
Linear algebra
in which the specific properties of vector spaces are studied (including matrices)
Universal algebra
in which properties common to all algebraic structures are studied
Algebraic number theory
in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
Algebraic geometry
applies abstract algebra to the problems of geometry
Algebraic combinatorics
in which abstract algebraic methods are used to study combinatorial questions.
Variables
symbols that denote numbers
The purpose of using variables
symbols that denote numbers, is to allow the making of generalizations in mathematics
Expressions
may contain numbers, variables and arithmetical operations. These are conventionally written with ‘higher-power’ terms on the left
Addition
Is Written as a + b
commutative law of Addition
a + b = b + a
associative law of addition
(a + b) + c = a + (b + c)
identity element of addition
0, which preserves numbers: a + 0 = a
inverse operation of addition
Subtraction ( – )
Multiplication
Is Written as a × b or a•b
commutative law of Multiplication
a × b = b × a
Associative law of Multiplication
(a × b) × c = a × (b × c)
Identity element of Multiplication
1, which preserves numbers: a × 1 = a
inverse operation of Multiplication
Division ( / )
Exponentiation
Is Written as ab or a^b
commutative law of Exponentiation
Not commutative a^b≠b^a
Associative law of Exponentiation
Not associative
identity element of Exponentiation
1, which preserves numbers: a^1 = a
inverse operation of Exponentiation
Logarithm (Log)
The operation of addition
means repeated addition of ones: a + n = a + 1 + 1 +…+ 1 (n number of times)
The operation of addition
has an inverse operation called subtraction: (a + b) − b = a, which is the same as adding a negative number, a − b = a + (−b)
The operation of multiplication
means repeated addition: a × n = a + a +…+ a (n number of times)
The operation of multiplication
has an inverse operation called division which is defined for non-zero numbers: (ab)/b = a, which is the same as multiplying by a reciprocal, a/b = a(1/b)
The operation of multiplication
distributes over addition: (a + b)c = ac + bc
The operation of multiplication
is abbreviated by juxtaposition: a × b ≡ ab
The operation of exponentiation
means repeated multiplication: a^n = a × a ×…× a (n number of times)
The operation of exponentiation
has an inverse operation, called the logarithm: a^loga^b = b = log`a a^b
The operation of exponentiation
distributes over multiplication: (ab)^c = a^cb^c
The operation of exponentiation
can be written in terms of n-th roots: a^m/n ≡ (n√a)^m and thus even roots of negative numbers do not exist in the real number system.
The operation of exponentiation
has the property: a^ba^c = a^b+c
The operation of exponentiation
has the property: (a^b)^c = a^bc.
The operation of exponentiation
in general a^b ≠ b^a and (a^b)^c ≠ a^(b^c)
Order of Operations
In mathematics it is important that the value of an expression is always computed the same way. Therefore, it is necessary to compute the parts of an expression in a particular order, known as the
Order of Operations
parenthesis and other grouping symbols including brackets, absolute value symbols, and the fraction bar
exponents and roots
multiplication and division
addition and subtraction
Equations
is the claim that two expressions have the same value and are equal.
Identities
Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called
Conditional equations
are true for only some values of the involved variables: x2 − 1 = 4.
Equation Solving
The values of the variables which make the equation true are the solutions of the equation and can be found through
Properties of equality
The relation of equality (=) is…
reflexive: b = b;
symmetric: if a = b then b = a;
transitive: if a = b and b = c then a = c.
The relation of equality (=) has the property…
that if a = b and c = d then a + c = b + d and ac = bd;
that if a = b then a + c = b + c;
that if two symbols are equal, then one can be substituted for the other.
The relation of equality (=) is
reflexive: b = b;
symmetric: if a = b then b = a;
transitive: if a = b and b = c then a = c.
reflexive
b = b
symmetric
if a = b then b = a
transitive
if a = b and b = c then a = c
The relation of equality (=) has the property
that if a = b and c = d then a + c = b + d and ac = bd;
that if a = b then a + c = b + c;
that if two symbols are equal, then one can be substituted for the other.
Properties of inequality
The relation of inequality (<) has the property
of transivity: if a < b and b < c then a < c; that if a < b and c < d then a + c < b + d; that if a < b and c > 0 then ac < bc; that if a < b and c < 0 then bc < ac.
then a < c
if a < b and b < c
then a + c < b + d
if a < b and c < d
then ac < bc
if a < b and c > 0
then bc < ac
if a < b and c < 0
The simplest equations to solve
are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. For example:
The central technique to linear equations
is to add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.
Quadratic equations
can be expressed in the form ax^2 + bx + c = 0, where a is not zero (if it were zero, then the equation would not be quadratic but linear).
Quadratic equations can also be solved
using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling).
All quadratic equations
will have two solutions in the complex number system, but need not have any in the real number system.
exponential equation
is an equation of the form aX = b for a > 0, which has solution
when b > 0
Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain
logarithmic equation
is an equation of the form log`a^X = b for a > 0, which has solution
radical equation
is an equation of the form X^m/n = a, for m, n integers, which has solution
system of linear equations
two equations in two variables, it is often possible to find the solutions of both variables that satisfy both equations.
Elimination method
An example of solving a system of linear equations is by using the elimination method: Multiplying the terms in the second equation by 2: Adding the two equations together to get: which simplifies to Since the fact that x = 2 is known, it is then possible to deduce that y = 3 by either of the original two equations (by using 2 instead of x) The full solution to this problem is then Note that this is not the only way to solve this specific system; y could have been solved before x.
substitution
An equivalent for y can be deduced by using one of the two equations. Using the second equation: Subtracting 2x from each side of the equation: and multiplying by -1: Using this y value in the first equation in the original system: Adding 2 on each side of the equation: which simplifies to Using this value in one of the equations, the same solution as in the previous method is obtained. Note that this is not the only way to solve this specific system; in this case as well, y could have been solved before x.
an operation
is an action or procedure which produces a new value from one or more input values.
unary and binary
There are two common types of operations:
Unary operations
involve only one value, such as negation and trigonometric functions.
Binary operations
take two values, and include addition, subtraction, multiplication, division, and exponentiation.
The logical values true and false
can be combined using logic operations, such as and, or, and not.
Vectors
can be added and subtracted.
Rotations
can be combined using the function composition operation, performing the first rotation and then the second.
Operations on sets
include the binary operations union and intersection and the unary operation of complementation.
Operations on functions
include composition and convolution
Operations
may not be defined for every possible value.
domain
The values for which an operation is defined form a set called its
range
The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its
nonnegative numbers
the squaring operation only produces
nonnegative numbers
the codomain is the set of real numbers but the range is the
Operations can involve dissimilar objects
A vector can be multiplied by a scalar to form another vector
scalar
the inner product operation on two vectors produces a
operands, arguments, or inputs
The values combined are called
value, result, or output
the value produced is called
two inputs
Operations can have fewer or more than
An operation ω
is a function of the form ω : V → Y, where V ⊂ X1 × … × Xk.
The sets Xk
are called the domains of the operation
the set Y
is called the codomain of the operation
the fixed non-negative integer k (the number of arguments)
is called the type or arity of the operation
has arity one
a unary operation
has arity two
a binary operation
nullary operation
An operation of arity zero is simply an element of the codomain Y, called a
k-ary operation
An operation of arity k is called a
(k+1)-ary relation that is functional on its first k domains
k-ary operation is a
finitary operation
referring to the finite number of arguments (the value k)
operation
implies that the domain of the function is a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain)
operation
is synonymous with function, map and mapping, that is, a relation, for which each element of the domain (input set) is associated with exactly one element of the codomain (set of possible outputs).
equation
a mathematical statement that asserts the equality of two expressions, this is written by placing the expressions on either side of an equals sign (=).
Equations
often express relationships between given quantities, the knowns, and quantities yet to be determined, the unknowns.
unknowns
are denoted by letters at the end of the alphabet, x, y, z, w, …
knowns
are denoted by letters at the beginning, a, b, c, d, …
Solving the Equation
The process of expressing the unknowns in terms of the knowns is called
a solution or root of the equation
In an equation with a single unknown, a value of that unknown for which the equation is true is called
solution to the system
is an assignment of values to all the unknowns so that all of the equations are true. also called set simultaneous equations.
identity
a distinction is made between the equality sign ( = ) for an equation and the equivalence symbol () for an
constants
Letters from the beginning of the alphabet like a, b, c… often denote
variables
letters from the end of the alphabet, like …x, y, z, are usually reserved for the
Any real number can be added to both sides.
Any real number can be subtracted from both sides.
Any real number can be multiplied to both sides.
Any non-zero real number can divide both sides.
Some functions can be applied to both sides.
If an equation in algebra is known to be true, the following operations may be used to produce another true equation:
reflexive relation
is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be “is equal to”.
An irreflexive, or anti-reflexive
a binary relation R over a set X is symmetric
if it holds for all a and b in X that if a is related to b then b is related to a.
real number
a value that represents a quantity along a continuum, such as -5 (an integer), 4/3 (a rational number that is not an integer), 8.6 (a rational number given by a finite decimal representation), √2 (the square root of two, an algebraic number that is not rational) and π (3.1415926535…, a transcendental number).
number line or real line
Real numbers can be thought of as points on an infinitely long line where the points corresponding to integers are equally spaced called the
The real number system
can be defined axiomatically up to an isomorphism
change of variables
is a basic technique used to simplify problems in which the original variables are replaced with new ones; the new and old variables being related in some specified way.
difference of two squares, or the difference of perfect squares
is a squared (multiplied by itself) number subtracted from another squared number. It refers to the identity
the method of equating the coefficients
is a way of solving a functional equation of two polynomials for a number of unknown parameters. It relies on the fact that two polynomials are identical precisely when all corresponding coefficients are equal. The method is used to bring formulas into a desired form.