algebra encyclopedia

8/7/2019 Algebra Encyclopedia

1/17

Definition ofArithmetic Sequences and Series

A sequence is an ordered list of numbers and the sum of the terms of a sequence is a series. In an

arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant. The constant is

called the common difference (d). The formula for finding any term of an arithmetic sequence

is where a is the first term of the sequence, dis the common difference, and n is

the number of the term to find. To find the sum of a certain number of terms of an arithmetic sequence

use where Sn is the sum ofn terms, a is the first term, and an is the nth term. An

arithmetic sequence graphs as points along a line.

Definition ofAsymptotes

An asymptote is a l ine that the graph of a function approaches, but never intersects. An asymptote can

occur when a denominator in a function includes a variable that cannot be canceled out by something in

the numerator. Horizontal asymptotes are horizontal lines that the graph of a function approaches

as xtends to plus or minus infinity. Vertical asymptotes are vertical lines near which the function grows

to infinity. When a linear asymptote is not horizontal or vertical, it is called an oblique or slant

asymptote. This type of asymptote occurs when the numerator of a rational function is exactly one

degree greater than the denominator. Determining the asymptotes of a function is an important step in

sketching its graph.

Definition ofBinomial Theorem


2/17

The binomial theorem is used to expand binomial expressions (a + b) raised to any given power without

direct multiplication. For example: Starting with the first term

and progressing to the last, the exponent ofa decreases by one while the exponent ofb increases by

one, and the sum of the exponents ofa and b in each term isn. The binomial coefficients form a pattern

of 1 3 3 1. The coefficients correspond with the rows of Pascal's triangle and are determined by a

formula involving factorials. These observations form the basis of the binomial theorem in its expanded

form:

Using sigma notation and factorials, the binomial theorem becomes:

where the notation is a combination such as nCk(read "n choose

k"):

Definition ofCombinations

Combinations are used when solving counting problems involving groups of objects. For example, four

students need to be selected from a group of 35 students to host a student forum. How many

combinations of four students are possible? Think about combination problems as arrangements

ofrobjects, without regard to order and without repetition, selected from n distinct objects. In other

words, a combination ofn objects selected rat a time. The number of such combinations is denoted by:

This formula is referred to as "n choose r" (such as "35 choose 4") and is also known as the binomial

coefficient.

Definition ofCompleting the Square

Completing the Square is a technique used to solve quadratic equations, graph quadratic functions, and

evaluate integrals. This technique can be used when factoring a quadratic equation does not work or to

find irrational and complex roots. Completing the Square converts a quadratic

polynomial that is not a perfect square to a perfect square polynomial plus a

constant. For example, can be converted to . To solve for the quadratic

equation, , complete the square by taking half the coefficient ofxin the

quadratic and adding its square , or from our


3/17

example, . The general formula for Completing the Square is written

as where .

Definition ofCompound Inequalities

A compound inequality is an equation with two or more inequalities joined together with either "and" or

"or" (for example, and ; or ). When two inequalities are joined

with and, they are often written simply as a double inequality, like: . The solution of an

"and" inequality is the intersection of each individual inequality in the sentence. To find the intersection,

determine where the two graphs of the sentence overlap. The solution of an "or"inequality is the union

of each individual inequality in the sentence. In an "or" inequality, solve the two inequalities separately

and combine the solutions together.

Definition ofCramer's Rule

Given a system of l inear equations, Cramer's Rule uses determinants of a matrix to solve for just one of

the variables without having to solve the whole system of equations. When using Cramer's Rule, first set

up and evaluate the determinants. For example, the system:

and , has the unique solution

where ,

,

After solving for each determinant, substitute the value into the formula to find the solution to the

system of equations. Cramer's Rule can be generalized to systems of linear equations with more than

two variables by the formula , ,

Definition ofCubic Equations

Equations of the third degree are called cubic equations. The general form of a cubic

is . Cubic equations have three possible values for x, at least one of which is

real. There are several methods for solving cubic equations: factorization, rational root theorem,

Descartes rule of signs, Vieta's root theorem, and Cardano's method. Cardano's method is used for the

cubic equations that are not easy to solve. To use the method, substitute to eliminate


4/17

the x term. The new equation is where and . Once

the cubic is reduced to this form (called the depressed form), the steps for finding the roots are different

depending on the different solutions (all roots are real, some complex, neither is zero, etc.).

Definition ofDeterminant

A determinant is a scalar number that is calculated from a square matrix. A determinant is differentfrom a matrix in that a determinant has a numerical value, whereas a matrix does not. Determinants

play an important role in finding the inverse of a matrix and determining whether systems of linear

equations are solvable. There are two ways to write a determinant: "detA" or . The determinant of

a 22 matrix is the product of the elements on the main diagonal minus the product of the elements off

the main diagonal.

The determinant of a 33 matrix is:

There are numerous formulas for calculating determinants for larger matrices.

Definition ofDiscriminants

The discriminant of an equation gives an idea of the number of roots and the nature of roots of the

equation. In other words, it "discriminates" between the possible solutions. The discriminant is the

expression found under the square root part of the quadratic formula (that

is, . The value of tells how many solutions, roots, or x-intercepts the

quadratic equation will have.

y If , there are two real solutions.

y If , there is one real solution.

y If , there are no real solutions, but there are two complex imaginarysolutions.


5/17

To find the solutions, manipulate the quadratic equation to standard form ( ),

determine a, b, and c, and plug those values into the discriminant formula.

Definition ofEllipses Equations

An ellipse is the figure consisting of all points for which the sum of their distances to two fixed points

(called the foci) is a constant. In the coordinate plane, an ellipse is the figure consisting of all points in

the plane whose Cartesian coordinates satisfy the equations:

Horizontal:

Vertical:

a = horizontal "radius"

b = vertical "radius"

c = distance of center to focus

Horizontal (pictured):

Vertical:

Where:

1. The center of the ellipse is (h, k).2. The length of the horizontal axis is 2a.3. The length of the vertical axis is 2b.4. Ifa > b, the ellipse is horizontal; ifb > a, the ellipse is vertical.

Definition ofEquations of Conics

5. A conic is the intersection of a plane and a cone. Changing the angle and location ofintersection can produce a circle, ellipse, parabola, or hyperbola. The equation of every

conic is written in the form: . When rewritten

in the form , this equation yields each conic

section as the intersection of the graph of the quadratic form


6/17

and the plane . Changing coordinates yields equations for each

conic in standard form:

Circle:

Ellipse:

Parabola:

Hyperbola:

Definition ofExponential Equations

An exponential equation is one in which a variable occurs in the exponent, for

example, . When both sides of the equation have the same base, the exponents on

either side are equal by the property if , then . Important logarithmic rules used

to solve exponential equations include:

Exponential equations are also solved using logs, either common (log) or natural (ln). These

types of logs can be expressed as:

The natural logarithm of a numberx (written as ln(x)) is the power to which ewould have to be

raised to equal x, where e is an irrational number approximately equal to 2.718281828. When the

base is not the same, use the Change-of-Base formula:

Definition ofExponential Functions

Exponential functions are functions of the form for a fixed base b that could

be any positive real number. Exponential functions are often associated with exponential growth, which

means that that there is a consistent fixed period over which the function will double (or triple, or

quadruple, etc.) and the change is always a fixed proportion.

The graph of is upward sloping, and increases faster as xincreases.


7/17

The main rules used in manipulating exponential functions are:

The natural exponential function is where e = 2.718281828 (approximately). This function

is distinguished among all exponential functions because its rate of growth at xis equal to the value exof

the function at x

Definition ofFactor Theorem

The factor theorem is a theorem linking factors and zeros of a polynomial. It is commonly applied to

factorizing and finding the roots of polynomial equations. The theorem states that isa factor of a

polynomial f(x)if ;that is,ris a root off(x). Factor theorem problems are typically solved by

applying synthetic division and then checking for a zero remainder. The remainder theorem is often

used with the factor theorem. It states that if a polynomial f(x) is divided by a linear divisor , the

remainder is f(r). If , then the remainder is 0 and , showing

that is a factor off(x).

Definition ofFactorials

Factorials are often used in algebra, combinatorics, and calculus, in particular, for solving permutation

and combination problems. Factorials are products indicated by an exclamation point. For instance, "five

factorial" is written as "5!" and means . In general, n! is the product of

descending natural numbers where . An important fact to

remember is that the factorial of 0 is 1 (that is, ). There are also factorials of real and complex

numbers using other formulas such as the gamma function.

Definition ofFactoring Polynomials

Factoring polynomials involves breaking up a polynomial into simpler terms (the factors) such that when

the terms are multiplied together they equal the original polynomial. Factoring helps solve complex

equations so they are easier to work with. Factoring polynomials includes:

y Finding the greatest common factory Grouping like terms


8/17

y Finding the difference of perfect squares:

y Finding the sum or difference of two perfect cubes:;

y Factoring quadratic trinomials:y Factoring polynomials with degree greater than 2:y Factoring rationals using GCD and

LCM:

y Factoring real numbers:y Factoring complex numbers:

Definition ofFactoring Trinomials

Factoring trinomials means finding two binomials that when multiplied together produce the given

trinomial. Trinomials take many forms, but basically use the same methods for factoring. Some

examples are difference of squares, perfect square trinomial, or trial and error. Always look for the

greatest common factor before factoring any trinomial.

y For trinomials of the form or , find the factors for the firstposition, then the factors for the last position such that their product equals c (the constant) and

at the same time their sum equals b.

y Perfect Square Trinomial: or

y Factoring a Difference of Two Squares:

Definition ofFinding Zeros

A zero or root of a polynomial function is a number that, when plugged in for the variable,

makes the function equal to zero. To find all the zeros of a polynomial function and the possible

rational roots of a polynomial equation, use the rational zero theorem. The theorem states that a

polynomial with integer coefficients has possible rational zeros equal to the factors of the

constant termp divided by the factors of the leading coefficient q: . Not every

number in the list of resulting possible factors is a zero of the function, but every rational zero of


9/17

the polynomial function will appear somewhere in this list. Once the possible zeros are found,

use synthetic division to test those zeros.

Definition ofFundamental Counting Principle

If Task A can be performed in any one ofn1 ways, and, if after Task A is performed, Task B can be

performed in any one ofn2 ways, then the combination of tasks, A followed by B, can be performed

in ways. More generally, suppose a task involves a sequence ofkchoices. Let n1 be the number

of ways the first stage can occur and n2 be the number of ways the second stage can occur after the first

stage has occurred. Continuing in this way, let nkbe the number of ways the kth stage can occur after

the first k 1 stages or events have occurred. Then the total number of different ways the task can

occur is .

Definition ofFundamental Theorem of Algebra

The fundamental theorem of algebra states that every polynomial equation over the field of complex

numbers of degree higher than one has a complex solution. Polynomials of the

form , with a, b,... coefficients real or complex, can be factored completely

into where the r, s,... are complex numbers. The theorem also gives the number of

solutions to the polynomial equation. Iff(x) is a polynomial of degree n, then f(x) has exactly n roots,

counting complex numbers and multiple roots. For example, the polynomial is of degree

three, so it must have three roots (counting complex roots and multiple roots).

Definition ofGeometric Sequences and Series

In a geometric sequence, each term is equal to the previous term, multiplied (or divided by) a constant.

The constant is called the common ratio (r).To find the common ratio, divide the second term by the

first term. To find any term of a geometric sequence, use where a is the first term of the

sequence, ris the common ratio, and n is the number of the term to find. To find the sum of a certain

number of terms of a geometric sequence, use where Sn is the sum ofn terms, a is the

first term, and ris the common ratio. Geometric sequences graph as points along the graph of an

exponential function.

Definition ofGraphing Polynomial Functions

A polynomial function is a function that can be written in the

form , where are

real numbers and n is a nonnegative integer. Polynomial functions contain no discontinuities in their

behavior, have distinct slopes and features, and have end behaviors that approach infinity. These

functions are excellent for demonstrating real-life situations, such as trends. To graph a polynomial

function, follow these steps:

1. Determine the graph's end behavior by using the Leading Coefficient Test.


10/17

2. Find the x-intercepts or zeros of the function.3. Find the y-intercept of the function.4. Determine if there is any symmetry.5. Find the number of maximum turning points.6. Find extra points.7. Draw the graph.

The graph of polynomial functions is always a smooth continuous curve.

Definition ofGraphing Quadratic Functions

The graph of a quadratic function is a parabola. The standard form of a quadratic function is written

as . If , the parabola opens upward; if , it opens downward. To

graph the function, first find the vertex by:

1. Either completing the square to rewrite the function in the form

(the vertex is (h, k)) or finding the line of symmetry using the equation , which

calculates the x-coordinate of the vertex, and then finding the yvalue of the vertex by

substituting the value ofxinto the original equation.

2. Then find the x- and y-intercepts of the graph. To find the x-intercepts, solve the quadratic

equation, . To find the y-intercept of , find f(0).

Definition ofGraphing Systems of Equations

The graph of a system of two linear equations produces three scenarios:

1. The lines have one point of intersection. This is the only solution of the system.


11/17

2. The lines are parallel and there is no point that satisfies both equations. The system has nosolution.

3. The lines coincide and the equations have the same graph every solution of one equation is asolution of the other. There are an infinite number of solutions.

To solve systems of linear inequalities, graph each individual inequality, and then find the overlaps of

the various solutions. The solution region for a system of inequalities can be a bounded geometric figure

or an unbounded figure, which has solutions that go off forever in some direction.


12/17

Definition ofInverse and Direct

A direct function (also called an identity function) is a function that always returns the same value as its

argument. It is denoted by . On the coordinate plane, the graph of the direct function is

. Two functions, fand g, are inverses of each other when f[g(x)] and g[f(x)] equal x. The inverse

function is denoted by f1(x). The graph of an inverse function is reflected about the line

where , or where :

The function must pass the vertical and horizontal line test to have an inverse. To find the inverse of a

function, replace f(x) with y, interchange xwith y, and solve for y

Definition ofLinear Programming

Linear programming is often used in business to find maximum profit or minimum cost. The first step insolving linear programming problems is to set up a function that represents cost, profit, or some other

quantity to be maximized or minimized subject to the constraints of the problem. Then define the

constraints with a system of linear inequalities. The solution of these systems of inequalities is called the

feasible region. Graph the inequalities to determine the coordinates of the vertices of the region.

Evaluate the function at each vertex. The largest and smallest of those values are the maximum and

minimum values of the function, respectively.


13/17

Definition ofLogarithmic Equations

Logarithmic equations contain logarithmic expressions and constants. A logarithm is another way to

write an exponent and is defined by if and only if . When one side of the

equation contains a single logarithm and the other side contains a constant, the equation can be solved

by rewriting the equation as an equivalent exponential equation using the definition of logarithm from

above. For example, ; ; . If one side of a logarithmic equation

contains more than one logarithm, use the properties of logarithms to condense it into a single

logarithm. Properties of logarithms basically change multiplication into addition, division into

subtraction, exponent into multiplication, and radical into division.

Definition ofLogarithmic Properties

The logarithm base b of a number xis the power to which b must be raised in order to equal x. This is

written logb x. For instance, because . Logs have four basic properties:

1. Product Rule: The log of a product is equal to the sum of the log of the first base and the log ofthe second base ( ).

2. Quotient Rule: The log of a quotient is equal to the difference of the logs of the numerator anddenominator ( ).

3. Power Rule: The log of a power is equal to the power times the log of the base ().

4. Change of Base Formula: The log of a new base is the log of the new base divided by the log ofthe old base in the new base ( ).

Definition ofMatrix Operations

Basic operations on matrices:


14/17

y To add or subtract matrices they must have the same number of rows and columns.Addition: (to form a new matrix z, to element in row i,

column jof matrix A add element in row i, column jof matrix B)

Subtraction: (to element in row i, column jof matrix A deduct

element in row i, column jof matrix B)

y Multiplication/Division by a scalar: where c isthe constant (scalar)

y Multiplication: , if the number of columns in A = number of rows inB; (add the products obtained by multiplying

elements in each irow of matrix A by elements in each jcolumn of matrix i)

Definition ofPermutations

A permutation of a number of objects is the number of different ways the objects can be

arranged, keeping order in mind (that is, which is first, second, third, and so on). When choosing

some objects from a larger number of objects, the position of the chosen objects is also

important. Think about permutation problems as the order ofr objects selected from n distinct

objects, orn objects taken r at a time. The number of such permutations is denoted

by:

For example, six students enter a contest. In how many ways can the first three places be

determined?

Note that n! is the number of permutations (ways of arranging) exactly n things.

Definition ofPolynomial Functions

A polynomial function has the form ,

where are real numbers and n is a nonnegative integer. In other words, a polynomial

is the sum of one or more monomials with real coefficients and nonnegative integer exponents. The

degree of the polynomial function is the highest value for n where anis not equal to 0. Polynomial

functions of only one term are called monomials or power functions. A power function has the

form . For a polynomial function f, any number rfor which is called a zero

or root of the function f. When a polynomial function is completely factored, each of the factors helps

identify zeros of the function

Definition ofQuadratic Formula


15/17


16/17

The remainder theorem is based on synthetic division, which is the process of dividing a polynomial f(x)

by a polynomial D(x) and finding the remainder. This is written as ,

where f(x) is the dividend,Q(x) is the quotient, D(x) is the divisor, and R(x) is the remainder. The

remainder theorem takes this one step further and states that if a polynomial f(x) is divided by a linear

divisor , the remainder is f(a). The factor theorem is used often with the remainder theorem in

that, when dividing by a number that is a potential root of the polynomial and arriving at a zero

remainder in the synthetic division, the number is a root, and xminus the number is a factor.

Definition ofSecond-Degree Equations and Inequalities

Second-degree equations involve at least one variable that is squared, or raised to a power of two. One

of the most well-known second-degree equations is the quadratic where a,

b, and c are constants and a is not equal 0. Second-degree equations have two possible

solutions: and

The graph of a second-degree equation produces a parabola. The solutions to the equation representwhere the parabola crosses the x-axis. The general form of second-degree inequalities

is . Solving inequalities involves finding all possible values of the variable

that will make the inequality true.

Definition ofSolving Complex Equations

The nature of the roots of a quadratic equation will take three forms: two real roots, one real root

repeated, or two complex roots. When a quadratic equation with real-number coefficients has a

negative discriminant (the expression that appears under the square root (radical) sign in the quadratic

formula), then the two solutions of the equation are complex roots. That is, when , the

roots by the quadratic formula, and , form complex

conjugate pairs. A property of a complex conjugate pair is that their product is always a non-negative

real number. For example, two complex numbers and when multiplied becomes a

non-negative real number: .

Definition ofSolving Quadratic Equations

There are many ways to solve quadratic equations, such as factoring, taking square roots, completing

the square, and using the quadratic formula. Factoring involves setting each factor equal to zero and

solving each factor. When the quadratic has a squared part and a number part (for

example, ), take the square root. When the equation cannot be solved by

factoring and is not in the form ready for a square root, complete the square or use the quadratic

formula. Completing the square rearranges the quadratic into a squared part so it equals a number (for

example, , making it easier to solve. The quadratic formula


17/17

can always find the solution to a quadratic equation: for , the value ofxis given

by .

Definition ofSolving Systems of Equations

Four methods for solving systems of equations are:

1. Graph every equation in the system and then use the graph to find the coordinates of thepoint(s) where the graphs intersect. The point of intersection is the solution. This method is not

helpful when the solution is not integers.

2. Substitution: First solve one of the equations for a variable, and then substitute it into the otherequation(s).

3. Elimination: Add equations together to eliminate variables, often by multiplying each equationby a constant so that the resulting coefficient of one of the variables is 0. This method is

effective with systems of three or more variables.

4. Matrices: Create coefficient and constant matrices and solve with row reduction.

Definition ofSolving Systems of Nonlinear Equations

A nonlinear system of equations is a set of equations where one or more terms have a variable of

degree two or higher and/or there is a product of variables in one of the equations. Most real-life

physical systems are non-linear systems, such as the weather. Solving nonlinear systems of

equations is much the same as solving linear equations: use either the substitution method or

elimination. The main difference is that some of the solutions may be complex. When the

solutions are real, they will represent the coordinates of the points where the graphs of the two

functions intersect. Complex solutions never represent intersections of two curves.

Definition ofSubstitution Method

One way to solve systems of equations is by substitution. In this method, solve an equation for one

variable, then substitute that solution in the other equation, and solve. Use substitution as a method for

solving a system of equations when the number of equations and variables is equal (if two variables,

there must be two equations; three variables, three equations, etc.) and one of the equations can easily

be solved for one variable. If there is no solution to the system of equations (parallel lines), thesubstitution method will result in something nonsensical, such as . One of the strengths of the

substitution method is that it works for systems of equations that are difficult or impossible to graph.

algebra encyclopedia

Documents