# College Algebra

## 100 terms

### abscissa

The x-coordinate of a point in the coordinate plane.

### absolute value

The absolute value of a number x, denoted by |x| is its distance form zero on the number line.

In the set of real numbers, 0 is the additive identity element since the sum of any real number and 0 is that number.

In the set of real numbers, the additive inverse of any real number x is its opposite, -x, since x + (-x) = 0.

### arithmetic sequence

A sequence of numbers or terms in which each term after the first is obtained from the term that precedes it by adding the same constant, called the common difference. The nth term of an arithmetic sequence is a`n = a`1 + (n-1)d where a`1 is the first term and d is the common difference.

### arithmetic series

The indicated sum of the terms of an arithmetic sequence. The sum of the first n terms of an arithmetic sequence is S`n= n/2 (a`1 + a`n).

### associative law

The order in which three real numbers are added or multiplied does not matter.

### asymptote

A line that a graph approaches, but does not intersect, as x increases or decreases without bound.

### axis of symmetry

A line that divides a graph so that if the graph is folded on the line the two parts of the graph will exactly coincide.

### binomial

A polynomial with two unlike terms.

### binomial theorem

A formula that tells how to expand a binomial of the form (a+b)`n without performing repeated multiplications.

### change of base formula

The formula log`ac= log`bc/log`ba changes the base of log`ac from a to b.

### characteristic

The integer part of a common logarithm.

### circle

The set of points (x, y) in a plane that are the same distance r from a given point (h, k) called the center. Thus, (x-h)^2 + (y-k)^2 = r^2.

### closure property

A set of numbers is closed under an operation if the number that the operation produces is a member of the same set.

### coefficient

The numerical factor of a monomial.

### combination

A subset of a set of objects in which order does not matter.

### combination formula

The combination of n objects taken r at a time, denoted by `nC`r, is given by the formula `nC`r= `nP`r/r! = n!/r!(n-r)!

### common difference

In an arithmetic sequence, the constant that is added to each term after the first in order to obtain the next consecutive term.

### common logarithm

A logarithm whose base is 10.

### common ratio

In geometric sequence, the constant that multiplies each term after the first in order to obtain the next consecutive term.

### commutative law

The order in which two real numbers are added or multiplied does not matter.

### complex fraction

A fraction that contains other fractions in its numerator, denominator, or in both the numerator and denominator.

### complex number

A number that can be written in the form a + bi where a and be are real numbers and i = the sq. root of -1.

### composite function

The composite function f o g is the function that consists of the set of function values f(g(x)), provided g(x) is in the domain of function f.

### conic section

A curve formed with a plane intersects a right circular cone. Circles, ellipses, hyperbolas, and parabolas are conic sections.

### conjugate axis

The axis of symmetry of a hyperbola that is perpendicular to the transverse axis at its center. The length of the conjugate axis is denoted by 2b.

### conjugate pair

The sum and difference of the same two terms, as in a + sq. root of b and a - sq. root of b or a + bi and a - bi.

### consistent system

A system of linear equations that has a unique solution.

### constant

A quantity that is fixed in value. In the equation y = 3x, x and y are variables and 3 is a constant.

### coordinate plane

The region formed when a horizontal number line and a vertical number line intersect at their zero points.

### counting principle

If activity A can occur in m ways and activity B can occur in n ways, then both events can occur in m times n ways.

### Cramer's rule

A method that uses determinants to solve a system of linear equations.

### degree of a monomial

The sum of the exponents of the variable factors of the monomial.

### degree of a polynomial

The greatest degree of the monomial terms of the polynomial.

### determinant of a 2 x 2 matrix

If matrix A = [ a c b d ], then the determinant of A, denoted by |a|, is a d - b c/

### dependent variable

For a function of the form y = f(x), y is the dependent variable.

### dimension

The size of a matrix determined by the number of its rows and columns.

### direct variation

Variable y varies directly as x if y/x= k or (y = kx) where k is called the constant of variation.

### discriminant

For a quadratic equation ax^2 + bx + c = 0, the quantity b^2-4ac.

### distance formula

The distance between the points (x`1, y`1) and (x`2, y`2) is the sq. root of (x`2-x`1)^2 + (y`2-y`1)^2.

### distributive law

For any real numbers a, b, and c, a(b+c) = ab + ac.

### domain (relation)

The set of all possible first members of the ordered pairs that comprise a relation.

### domain (variable)

The set of all possible replacements for a variable.

### element

A member of a set of an entry in a matrix.

### ellipse

The set of all points P in a plane such that the sum of the distances from P to two fixed points, called foci, is constant.

### equilateral hyperbola

The hyperbola xy= k (k doesn't = 0) whose asympototes are the coordinate axes. Also called a rectangular hyperbola.

### equivalent equations

Equations that have the same solution set. Thus, 2x = 6 and x = 3 are these.

### even function

Function f is even if f(-x) = f(x) for all x in the domain of f.

### exponent

In x^n, the number n is the exponent and is the number of times the base x is used as a factor in a product. Thus, x^3= x x x.

### exponential equation

An equation in which the variable appears in an exponent.

### exponential function

A function of the form y = b^x where b is a positive constant that is not equal to 1.

### factor

A number or variable that is being multiplied in a product.

### factor theorem

The binomial x - r is a factor of the polynomial P(x) is P(r) = 0.

### factoring

The process by which a number or polynomial is written as the product of two or more terms.

### factoring completely

Factoring a number or polynomial into its prime factors.

### factorial n

Denoted by n! and defined for any positive integer n as the product of consecutive integers from n to 1. Thus, 5! = 5 4 3 2 1 = 120. By definition, 0! = 1.

### field

A set of numbers and two operations that are defined on the set such that there exist identity and inverse elements for each operation and the closure, commutative, associative, and distributive properties hold. The set of real numbers forms a field with respect to the operations of addition and multiplication.

### focus (foci)

A fixed point or points that help to determine the shape of an ellipse, hyperbola, or parabola.

### FOIL

The rule for multiplying two binomials horizontally.

### function

A relation in which no two ordered pairs have the same first member and different second members.

### fundamental theorem of algebra

A polynomial equation has at least one root, real or nonreal. An nth-degree polynomial equation has exactly n roots, provided each root is counted as many times as it occurs. The roots may be real or imaginary. If a polynomial equation with real coefficients has imaginary roots, then they occur in conjugate pairs.

### geometric sequence

A sequence of numbers or terms in which each term after the first is obtained by multiplying the term that precedes it by the same constant, called the common ratio. The nth term of a geometric sequence is a`n = a`1r^(n-1), where a`1 is the first term and r is the common ratio.

### geometric series

The indicated sum of the terms of a geometric sequence. The sum of the first n terms of a geometric sequence is S`n = a`1(1-r^n)/1-r

### greatest common factor (GCF)

This of two or more monomials is the monomial with the greatest coefficient and the variable factors of the greatest degree that are common to all the given monomials. The GCF of 8a^2b and 20ab^2 is 4ab.

### greatest integer function

The function f(x) = [x] whose function value is the largest integer that is less than or equal to x.

### horizontal line test

If no horizontal line intersects a graph in more than one point, the graph represents a one-to-one function.

### hyperbola

The set of all points P in a plane such that the absolute value of the difference of the distances from P to two fixed points, called foci, is constant.

### identity

An equation that is true for all possible replacements of the variable.

### imaginary unit

The number denoted by i where i = the sq. root of -1.

### imaginary number

A number of the form bi where b is a nonzero real number and i is the imaginary unit.

### inconsistent system

A system of linear equations that has no solution.

### independent variable

For a function of the form y = f(x), x is the independent variable.

### index

The number k in the radical expression ksq.root of x is the root of x to be taken. In a square root radical the index is omitted and is understood to be 2.

### inequality

A sentence that expresses an inequality relation by using a symbol such as < (is less than), <(w/ line under; is less than or equal to), > (is greater than), > (with line under; is greater than or equal to), or =(with line through; is unequal to).

### infinite sequence

A sequence whose domain (set of subscripts) is the set of natural numbers.

### infinite series

The indicated sum of the terms of a infinite sequence.

### integer

A number from the set [..., -3, -2, -1, 0, 1, 2, 3, ...].

### intermediate value theorem

If P(x) is a polynomial with real coefficients and P(a) and P(b) have opposite signs, then a real root of P(x)= 0 lies between a and b.

### inverse of a function

The relation obtained by interchanging the first and second members of each ordered pair of the function. The result may or may not be a function.

### inverse of a matrix

The multiplicative inverse of a square matrix A, if it exists, is the matrix A^-1 such that AA^-1 = A^-1A= I, where I is the identity matrix.

### inverse variation

If y varies inversely as x, then xy = k (or y = k/x) where k is a nonzero number called the constant of variation.

### irrational number

A number that cannot be expressed as the quotient of two integers.

For a polynomial in one variable, the number that multiplies the variable with the greatest exponent.

### linear equation

An equation in which the greatest exponent of any variable is 1.

### logarithmic function

The inverse of the exponential function.

### logarithm of x

An exponent that represents the power to which a given base must be raised to produce a positive number x.

### major axis

The longer of the two axes of symmetry of an ellipse whose endpoints, called vertices, are points on the ellipse. The length of the major axis is denoted by 2a.

### mantissa

The decimal part of a common logarithm.

### mapping

A pairing of each member of a set with exactly one member of another set.

### mathematical induction

A special method of proof used to prove or disprove formula type of statements that are expressed in terms of a positive integer n.

### matrix

A rectangular array of numbers or terms called elements. An m x n matrix has m rows and n columns. The element in the ith row and jth column of matrix A is named a`ij.

If A and B are m x n matrices, then A + B is an m x n matrix in which each element is the sum of the elements in the corresponding positions of A and B.

### matrix multiplication

If A is an m x n matrix and B is an n x p matrix, then AB is an m x p matrix whose entry in the ith row and jth column is the sum of the products obtained by multiplying the elements in the ith row of matrix A by the corresponding elements in the jth column of matrix B.

### midpoint formula

The midpoint of the line segment whose endpoints are (x`1, y`1) and (x`2, y`2) is (x`1 + x`2/2, y`1 + y`2/2).

### minor axis

The axis of symmetry of an ellipse that is perpendicular to the major axis at the center and whose endpoints are on the ellipse. The length of the minor axis is denoted by 2b.

### monomial

A number, variable, or the product of a number and a variable.

### multiplicative identity

In the set of real numbers, 1 is the multiplicative identity element since the product of any real number and 1 is that number.

### multiplicative identity matrix

For any square matrix A, the square matrix I that has 1 along its main diagonal and 0 everywhere else since AI = A.

### multiplicative inverse

In the set of real numbers, the multiplicative inverse of a nonzero number is the reciprocal of that numbers since their product is 1.