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Mathematics LibreTexts

Glossary

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absolute value | The distance from the graph of a number a to zero on a number line, denoted |a|.

absolute value function | The function defined by f(x)=|x|.

AC method | Method used for factoring trinomials by replacing the middle term with two terms that allow us to factor the resulting four-term polynomial by grouping.

addition property of equations | If A, B, C, and D are algebraic expressions, where A = B and C = D, then A + C = B + D.

addition property of equations | If A, B, C, and D are algebraic expressions, where A = B and C = D, then A + C = B + D.

algebraic expressions | Combinations of variables and numbers along with mathematical operations used to generalize specific arithmetic operations.

argument of the absolute value | The number or expression inside the absolute value.

argument of the absolute value | The number or expression inside the absolute value.

argument of the function | The value or algebraic expression used as input when using function notation.

arithmetic means | The terms between given terms of an arithmetic sequence.

arithmetic progression | Used when referring to an arithmetic sequence.

arithmetic series | The sum of the terms of an arithmetic sequence.

arithmetic series | The sum of the terms of an arithmetic sequence.

augmented matrix | The coefficient matrix with the column of constants included.

augmented matrix | The coefficient matrix with the column of constants included.

average cost | The total cost divided by the number of units produced, which can be represented by C(x)=C(x)x, where C(x) is a cost function.

average cost | The total cost divided by the number of units produced, which can be represented by C(x)=C(x)x, where C(x) is a cost function.

axis of symmetry | A term used when referencing the line of symmetry.

axis of symmetry | A term used when referencing the line of symmetry.

Back substitute | Once a value is found for a variable, substitute it back into one of the original equations, or its equivalent, to determine the corresponding value of the other variable.

Back substitute | Once a value is found for a variable, substitute it back into one of the original equations, or its equivalent, to determine the corresponding value of the other variable.

binomial | Polynomial with two terms.

binomial coefficient | An integer that is calculated using the formula: (nk)=n!k!(nk)!.

binomial coefficient | An integer that is calculated using the formula: (nk)=n!k!(nk)!.

binomial theorem | Describes the algebraic expansion of binomials raised to powers: (x+y)n=Σk=0n(nk)xnkyk.

binomial theorem | Describes the algebraic expansion of binomials raised to powers: (x+y)n=Σk=0n(nk)xnkyk.

breakeven point | The point at which profit is neither negative nor positive; profit is equal to zero.

breakeven point | The point at which profit is neither negative nor positive; profit is equal to zero.

Cartesian coordinate system | Term used in honor of René Descartes when referring to the rectangular coordinate system.

Cartesian coordinate system | Term used in honor of René Descartes when referring to the rectangular coordinate system.

change of base formula | logax=logbxlogba; we can write any base-a logarithm in terms of base-b logarithms using this formula.

change of base formula | logax=logbxlogba; we can write any base-a logarithm in terms of base-b logarithms using this formula.

circle in general form | The equation of a circle written in the form x2+y2+cx+dy+e=0.

circle in general form | The equation of a circle written in the form x2+y2+cx+dy+e=0.

circle in standard form | The equation of a circle written in the form (xh)2+(yk)2=r2 where (h,k) is the center and r is the radius.

circle in standard form | The equation of a circle written in the form (xh)2+(yk)2=r2 where (h,k) is the center and r is the radius.

co-vertices | Points on the ellipse that mark the endpoints of the minor axis.

co-vertices | Points on the ellipse that mark the endpoints of the minor axis.

codomain | Used when referencing the range.

codomain | Used when referencing the range.

coefficient matrix | The matrix of coefficients of a linear system in standard form written as they appear lined up without the variables or operations.

coefficient matrix | The matrix of coefficients of a linear system in standard form written as they appear lined up without the variables or operations.

combining like terms | Adding or subtracting like terms within an algebraic expression to obtain a single term with the same variable part.

combining like terms | Adding or subtracting like terms within an algebraic expression to obtain a single term with the same variable part.

common denominator | A denominator that is shared by more than one fraction.

common denominator | A denominator that is shared by more than one fraction.

common difference | The constant d that is obtained from subtracting any two successive terms of an arithmetic sequence; anan1=d.

common difference | The constant d that is obtained from subtracting any two successive terms of an arithmetic sequence; anan1=d.

common factor | A factor that is shared by more than one real number.

common factor | A factor that is shared by more than one real number.

common logarithm | The logarithm base 10, denoted logx.

common logarithm | The logarithm base 10, denoted logx.

common ratio | The constant r that is obtained from dividing any two successive terms of a geometric sequence; anan1=r.

common ratio | The constant r that is obtained from dividing any two successive terms of a geometric sequence; anan1=r.

completely factored | A polynomial that is prime or written as a product of prime polynomials.

completely factored | A polynomial that is prime or written as a product of prime polynomials.

completing the square | The process of rewriting a quadratic equation to be in the form (xp)2=q.

completing the square | The process of rewriting a quadratic equation to be in the form (xp)2=q.

complex conjugate | Two complex numbers whose real parts are the same and imaginary parts are opposite. If given a+bi, then its complex conjugate is abi.

complex conjugate | Two complex numbers whose real parts are the same and imaginary parts are opposite. If given a+bi, then its complex conjugate is abi.

complex rational expression | A rational expression that contains one or more rational expressions in the numerator or denominator or both.

complex rational expression | A rational expression that contains one or more rational expressions in the numerator or denominator or both.

composition operator | The open dot used to indicate the function composition (fg)(x)=f(g(x)).

composition operator | The open dot used to indicate the function composition (fg)(x)=f(g(x)).

compound inequalities | Two or more inequalities in one statement joined by the word “and” or by the word “or.”

compound inequalities | Two or more inequalities in one statement joined by the word “and” or by the word “or.”

compound interest formula | A formula that gives the amount accumulated by earning interest on principal and interest over time: A(t)=P(1+rn)nt.

compound interest formula | A formula that gives the amount accumulated by earning interest on principal and interest over time: A(t)=P(1+rn)nt.

conic section | A curve obtained from the intersection of a right circular cone and a plane.

conic section | A curve obtained from the intersection of a right circular cone and a plane.

conjugate axis | A line segment through the center of a hyperbola that is perpendicular to the transverse axis.

conjugate axis | A line segment through the center of a hyperbola that is perpendicular to the transverse axis.

conjugate binomials | The binomials (a+b) and (ab).

conjugate binomials | The binomials (a+b) and (ab).

conjugates | The factors (a+b) and (ab) are conjugates.

conjugates | The factors (a+b) and (ab) are conjugates.

constant function | Any function of the form f(x)=c where c is a real number.

constant function | Any function of the form f(x)=c where c is a real number.

constant of proportionality | Used when referring to the constant of variation.

constant of proportionality | Used when referring to the constant of variation.

constant of variation | The nonzero multiple k, when quantities vary directly or inversely.

constant of variation | The nonzero multiple k, when quantities vary directly or inversely.

constant polynomial | A polynomial with degree 0.

constant term | A term written without a variable factor.

constant term | A term written without a variable factor.

continuously compounding interest formula | A formula that gives the amount accumulated by earning continuously compounded interest: A(t)=Pert.

continuously compounding interest formula | A formula that gives the amount accumulated by earning continuously compounded interest: A(t)=Pert.

contradiction | An equation that is never true and has no solution.

contradiction | An equation that is never true and has no solution.

convergent geometric series | An infinite geometric series where |r|<1 whose sum is given by the formula: S=a11r.

convergent geometric series | An infinite geometric series where |r|<1 whose sum is given by the formula: S=a11r.

cost function | A function that models the cost of producing a number of units.

cost function | A function that models the cost of producing a number of units.

Cramer’s rule | The solution to an independent system of linear equations expressed in terms of determinants.

Cramer’s rule | The solution to an independent system of linear equations expressed in terms of determinants.

critical numbers | The values in the domain of a function that separate regions that produce positive or negative results.

critical numbers | The values in the domain of a function that separate regions that produce positive or negative results.

cross multiplication | If ab=cd then ad=bc.

cross multiplication | If ab=cd then ad=bc.

cube | The result when the exponent of any real number is 3.

cube | The result when the exponent of any real number is 3.

cube root function | The function defined by f(x)=x3.

cube root function | The function defined by f(x)=x3.

cubing function | The cubic function defined by f(x)=x3.

cubing function | The cubic function defined by f(x)=x3.

degree of a polynomial | The largest degree of all of its terms.

degree of a polynomial | The largest degree of all of its terms.

degree of a term | The exponent of the variable. If there is more than one variable in the term, the degree of the term is the sum their exponents.

degree of a term | The exponent of the variable. If there is more than one variable in the term, the degree of the term is the sum their exponents.

dependent system | A linear system with two variables that consists of equivalent equations. It has infinitely many ordered pair solutions, denoted by (x,mx+b).

dependent system | A linear system with two variables that consists of equivalent equations. It has infinitely many ordered pair solutions, denoted by (x,mx+b).

dependent variable | The variable whose value is determined by the value of the independent variable. Usually we think of the y-value of an ordered pair (x, y) as the dependent variable.

dependent variable | The variable whose value is determined by the value of the independent variable. Usually we think of the y-value of an ordered pair (x, y) as the dependent variable.

determinant | A real number associated with a square matrix.

determinant | A real number associated with a square matrix.

diameter | The length of a line segment passing through the center of a circle whose endpoints are on the circle.

diameter | The length of a line segment passing through the center of a circle whose endpoints are on the circle.

difference | The result of subtracting.

difference | The result of subtracting.

difference of cubes | a3b3=(ab)(a2+ab+b2), where a and b represent algebraic expressions.

difference of cubes | a3b3=(ab)(a2+ab+b2), where a and b represent algebraic expressions.

difference of squares | The special product obtained by multiplying conjugate binomials (a+b)(ab)=a2b2.

difference of squares | The special product obtained by multiplying conjugate binomials (a+b)(ab)=a2b2.

difference of squares | a2b2=(a+b)(ab), where a and b represent algebraic expressions.

difference of squares | a2b2=(a+b)(ab), where a and b represent algebraic expressions.

difference quotient | The mathematical quantity f(x+h)f(x)h, where h0, which represents the slope of a secant line through a function f.

difference quotient | The mathematical quantity f(x+h)f(x)h, where h0, which represents the slope of a secant line through a function f.

dilation | A non-rigid transformation, produced by multiplying functions by a nonzero real number, which appears to stretch the graph either vertically or horizontally.

dilation | A non-rigid transformation, produced by multiplying functions by a nonzero real number, which appears to stretch the graph either vertically or horizontally.

discriminant | The expression inside the radical of the quadratic formula, b24ac.

discriminant | The expression inside the radical of the quadratic formula, b24ac.

distance formula | Given two points (x1,y1) and (x2,y2), calculate the distance d between them using the formula d=(x2x1)2+(y2y1)2.

distance formula | Given two points (x1,y1) and (x2,y2), calculate the distance d between them using the formula d=(x2x1)2+(y2y1)2.

distance formula | Given two points (x1,y1) and (x2,y2), the distance d between them is given by d=(x2x1)2+(y2y1)2.

distance formula | Given two points (x1,y1) and (x2,y2), the distance d between them is given by d=(x2x1)2+(y2y1)2.

distributive property | Given any real numbers a, b, and c, a(b+c)=ab+ac or (b+c)a=ba+ca.

distributive property | Given any real numbers a, b, and c, a(b+c)=ab+ac or (b+c)a=ba+ca.

division | Divide functions as indicated by the notation: (f/g)(x)=f(x)g(x), where g(x)0.

division | Divide functions as indicated by the notation: (f/g)(x)=f(x)g(x), where g(x)0.

double root | A root that is repeated twice.

double root | A root that is repeated twice.

double-negative property | The opposite of a negative number is positive: −(−a) = a.

double-negative property | The opposite of a negative number is positive: −(−a) = a.

doubling time | The period of time it takes a quantity to double.

doubling time | The period of time it takes a quantity to double.

element | An object within a set.

element | An object within a set.

elementary row operations | Operations that can be performed to obtain equivalent linear systems.

elementary row operations | Operations that can be performed to obtain equivalent linear systems.

ellipse | The set of points in a plane whose distances from two fixed points have a sum that is equal to a positive constant.

ellipse | The set of points in a plane whose distances from two fixed points have a sum that is equal to a positive constant.

ellipse in general form | The equation of an ellipse written in the form px2+qy2+cx+dy+e=0 where p,q>0.

ellipse in general form | The equation of an ellipse written in the form px2+qy2+cx+dy+e=0 where p,q>0.

ellipse in standard form | The equation of an ellipse written in the form (xh)2a2+(yk)2b2=1. The center is (h,k) and the larger of a and b is the major radius and the smaller is the minor radius.

ellipse in standard form | The equation of an ellipse written in the form (xh)2a2+(yk)2b2=1. The center is (h,k) and the larger of a and b is the major radius and the smaller is the minor radius.

empty set | A subset with no elements, denoted Ø or { }.

empty set | A subset with no elements, denoted Ø or { }.

equivalent equations | Equations with the same solution set.

equivalent equations | Equations with the same solution set.

equivalent fractions | Two equal fractions expressed using different numerators and denominators.

equivalent fractions | Two equal fractions expressed using different numerators and denominators.

equivalent inequality | Inequalities that share the same solution set.

equivalent inequality | Inequalities that share the same solution set.

equivalent system | A system consisting of equivalent equations that share the same solution set.

equivalent system | A system consisting of equivalent equations that share the same solution set.

evaluating | The process of performing the operations of an algebraic expression for given values of the variables.

evaluating | The process of performing the operations of an algebraic expression for given values of the variables.

even integers | Integers that are divisible by 2.

even integers | Integers that are divisible by 2.

exponent | The positive integer n in the exponential notation an that indicates the number of times the base is used as a factor.

exponent | The positive integer n in the exponential notation an that indicates the number of times the base is used as a factor.

exponential form | An equivalent expression written using a rational exponent.

exponential form | An equivalent expression written using a rational exponent.

exponential function | Any function with a definition of the form f(x)=bx where b>0 and b1.

exponential function | Any function with a definition of the form f(x)=bx where b>0 and b1.

exponential growth/decay formula | A formula that models exponential growth or decay: P(t)=P0ekt.

exponential growth/decay formula | A formula that models exponential growth or decay: P(t)=P0ekt.

exponential notation | The compact notation an used when a factor a is repeated n times.

exponential notation | The compact notation an used when a factor a is repeated n times.

extracting the root | Applying the square root property as a means of solving a quadratic equation.

extracting the root | Applying the square root property as a means of solving a quadratic equation.

extraneous solutions | A solution that does not solve the original equation.

extraneous solutions | A solution that does not solve the original equation.

extraneous solutions | A properly found solution that does not solve the original equation.

extraneous solutions | A properly found solution that does not solve the original equation.

extrapolation | Using a linear function to estimate values that extend beyond the given data points.

extrapolation | Using a linear function to estimate values that extend beyond the given data points.

factorial | The product of all natural numbers less than or equal to a given natural number, denoted n!.

factorial | The product of all natural numbers less than or equal to a given natural number, denoted n!.

Factoring by grouping | A technique for factoring polynomials with four terms.

Factoring by grouping | A technique for factoring polynomials with four terms.

Factoring out the greatest common factor (GCF) | The process of rewriting a polynomial as a product using the GCF of all of its terms.

Factoring out the greatest common factor (GCF) | The process of rewriting a polynomial as a product using the GCF of all of its terms.

factors | Any of the numbers that form a product.

factors | Any of the numbers that form a product.

factors | Any of the numbers or expressions that form a product.

factors | Any of the numbers or expressions that form a product.

finite sequence | A sequence whose domain is {1,2,3,,k} where k is a natural number.

finite sequence | A sequence whose domain is {1,2,3,,k} where k is a natural number.

floor function | A term used when referring to the greatest integer function.

floor function | A term used when referring to the greatest integer function.

formulas | A reusable mathematical model using algebraic expressions to describe a common application.

formulas | A reusable mathematical model using algebraic expressions to describe a common application.

fraction | A rational number written as a quotient of two integers: ab, where b0.

fraction | A rational number written as a quotient of two integers: ab, where b0.

function | A relation where each element in the domain corresponds to exactly one element in the range.

function | A relation where each element in the domain corresponds to exactly one element in the range.

function notation | The notation f(x)=y, which reads “f of x is equal to y.” Given a function, y and f(x) can be used interchangeably.

function notation | The notation f(x)=y, which reads “f of x is equal to y.” Given a function, y and f(x) can be used interchangeably.

fundamental rectangle | The rectangle formed using the endpoints of a hyperbolas, transverse and conjugate axes.

fundamental rectangle | The rectangle formed using the endpoints of a hyperbolas, transverse and conjugate axes.

fundamental theorem of algebra | Guarantees that there will be as many (or fewer) roots to a polynomial function with one variable as its degree.

fundamental theorem of algebra | Guarantees that there will be as many (or fewer) roots to a polynomial function with one variable as its degree.

fundamental theorem of algebra | If multiple roots and complex roots are counted, then every polynomial with one variable will have as many roots as its degree.

fundamental theorem of algebra | If multiple roots and complex roots are counted, then every polynomial with one variable will have as many roots as its degree.

Gaussian elimination | Steps used to obtain an equivalent linear system in upper triangular form so that it can be solved using back substitution.

Gaussian elimination | Steps used to obtain an equivalent linear system in upper triangular form so that it can be solved using back substitution.

general term of a sequence | An equation that defines the nth term of a sequence commonly denoted using subscripts an.

general term of a sequence | An equation that defines the nth term of a sequence commonly denoted using subscripts an.

geometric means | The terms between given terms of a geometric sequence.

geometric means | The terms between given terms of a geometric sequence.

geometric progression | Used when referring to a geometric sequence.

geometric progression | Used when referring to a geometric sequence.

geometric series | The sum of the terms of a geometric sequence.

geometric series | The sum of the terms of a geometric sequence.

graph | A visual representation of a relation on a rectangular coordinate plane.

graph | A visual representation of a relation on a rectangular coordinate plane.

graph of the solution set | Solutions to an algebraic expression expressed on a number line.

graph of the solution set | Solutions to an algebraic expression expressed on a number line.

graphing method | A means of solving a system by graphing the equations on the same set of axes and determining where they intersect.

graphing method | A means of solving a system by graphing the equations on the same set of axes and determining where they intersect.

greatest common factor (GCF). | The largest shared factor of any number of integers.

greatest common factor (GCF). | The largest shared factor of any number of integers.

greatest common monomial factor (GCF) | The product of the common variable factors and the GCF of the coefficients.

greatest common monomial factor (GCF) | The product of the common variable factors and the GCF of the coefficients.

greatest integer function | The function that assigns any real number x to the greatest integer less than or equal to x denoted f(x)=[[x]].

greatest integer function | The function that assigns any real number x to the greatest integer less than or equal to x denoted f(x)=[[x]].

grouping symbols | Parentheses, brackets, braces, and the fraction bar are the common symbols used to group expressions and mathematical operations within a computation.

grouping symbols | Parentheses, brackets, braces, and the fraction bar are the common symbols used to group expressions and mathematical operations within a computation.

half-life | The period of time it takes a quantity to decay to one-half of the initial amount.

half-life | The period of time it takes a quantity to decay to one-half of the initial amount.

horizontal asymptote | A horizontal line to which a graph becomes infinitely close where the x-values tend toward ±∞.

horizontal asymptote | A horizontal line to which a graph becomes infinitely close where the x-values tend toward ±∞.

horizontal line test | If a horizontal line intersects the graph of a function more than once, then it is not one-to-one.

horizontal line test | If a horizontal line intersects the graph of a function more than once, then it is not one-to-one.

horizontal translation | A rigid transformation that shifts a graph left or right.

horizontal translation | A rigid transformation that shifts a graph left or right.

hyperbola | The set of points in a plane whose distances from two fixed points, called foci, has an absolute difference that is equal to a positive constant.

hyperbola | The set of points in a plane whose distances from two fixed points, called foci, has an absolute difference that is equal to a positive constant.

hyperbola in general form | The equation of a hyperbola written in the form px2qy2+cx+dy+e=0 or qy2px2cx+dy+e=0 where p,q>0.

hyperbola in general form | The equation of a hyperbola written in the form px2qy2+cx+dy+e=0 or qy2px2cx+dy+e=0 where p,q>0.

hyperbola opening left and right in standard form | The equation of a hyperbola written in the form (xh)2a2(yk)2b2=1. The center is (h,k), a defines the transverse axis, and b defines the conjugate axis.

hyperbola opening left and right in standard form | The equation of a hyperbola written in the form (xh)2a2(yk)2b2=1. The center is (h,k), a defines the transverse axis, and b defines the conjugate axis.

hyperbola opening upward and downward in standard form | The equation of a hyperbola written in the form (yk)2b2(xh)2a2=1. The center is (h,k), b defines the transverse axis, and a defines the conjugate axis.

hyperbola opening upward and downward in standard form | The equation of a hyperbola written in the form (yk)2b2(xh)2a2=1. The center is (h,k), b defines the transverse axis, and a defines the conjugate axis.

identity function | The linear function defined by f(x)=x.

identity function | The linear function defined by f(x)=x.

imaginary number | A square root of any negative real number.

imaginary number | A square root of any negative real number.

imaginary part | The real number b of a complex number a+bi.

imaginary part | The real number b of a complex number a+bi.

imaginary unit | Defined as i=1 where i2=1.

imaginary unit | Defined as i=1 where i2=1.

inclusive inequalities | Use the symbol to express quantities that are “less than or equal to” and for quantities that are “greater than or equal to” each other.

inclusive inequalities | Use the symbol to express quantities that are “less than or equal to” and for quantities that are “greater than or equal to” each other.

inconsistent system | A system with no simultaneous solution.

inconsistent system | A system with no simultaneous solution.

independent system | A linear system with two variables that has exactly one ordered pair solution.

independent system | A linear system with two variables that has exactly one ordered pair solution.

indeterminate | A quotient such as 00 is a quantity that is uncertain or ambiguous.

indeterminate | A quotient such as 00 is a quantity that is uncertain or ambiguous.

index | The positive integer n in the notation n that is used to indicate an nth root.

index | The positive integer n in the notation n that is used to indicate an nth root.

index | The positive integer n in the notation n that is used to indicate an nth root.

index | The positive integer n in the notation n that is used to indicate an nth root.

index of summation | The variable used in sigma notation to indicate the lower and upper bounds of the summation.

index of summation | The variable used in sigma notation to indicate the lower and upper bounds of the summation.

infinite sequence | A sequence whose domain is the set of natural numbers {1,2,3,}.

infinite sequence | A sequence whose domain is the set of natural numbers {1,2,3,}.

infinity | The symbol ∞ indicates the interval is unbounded to the right.

infinity | The symbol ∞ indicates the interval is unbounded to the right.

integers | The set of positive and negative whole numbers combined with zero: {…, −3, −2, −1, 0, 1, 2, 3, …}.

integers | The set of positive and negative whole numbers combined with zero: {…, −3, −2, −1, 0, 1, 2, 3, …}.

interpolation | Using a linear function to estimate a value between given data points.

interpolation | Using a linear function to estimate a value between given data points.

intersection | The set formed by the shared values of the individual solution sets that is indicated by the logical use of the word “and,” denoted with the symbol .

intersection | The set formed by the shared values of the individual solution sets that is indicated by the logical use of the word “and,” denoted with the symbol .

inverse properties of the logarithm | Given b>0 we have logbbx=x and blogbx=x when x>0.

inverse properties of the logarithm | Given b>0 we have logbbx=x and blogbx=x when x>0.

inversely proportional | Used when referring to inverse variation.

inversely proportional | Used when referring to inverse variation.

Irrational numbers | Numbers that cannot be written as a ratio of two integers.

Irrational numbers | Numbers that cannot be written as a ratio of two integers.

joint variation | Describes a quantity y that varies directly as the product of two other quantities x and z: y=kxz.

joint variation | Describes a quantity y that varies directly as the product of two other quantities x and z: y=kxz.

leading coefficient | The coefficient of the term with the largest degree.

least common denominator | The least common multiple of a set of denominators.

least common denominator | The least common multiple of a set of denominators.

linear equation with one variable | An equation that can be written in the standard form ax+b=0, where a and b are real numbers and a0.

linear equation with one variable | An equation that can be written in the standard form ax+b=0, where a and b are real numbers and a0.

linear function | Any function that can be written in the form f(x)=mx+b

linear function | Any function that can be written in the form f(x)=mx+b

linear inequality | Linear expressions related with the symbols , <, , and >.

linear inequality | Linear expressions related with the symbols , <, , and >.

linear inequality with two variables | An inequality relating linear expressions with two variables. The solution set is a region defining half of the plane.

linear inequality with two variables | An inequality relating linear expressions with two variables. The solution set is a region defining half of the plane.

linear systems | A set of two or more linear equations with the same variables.

linear systems | A set of two or more linear equations with the same variables.

logarithm base b | The exponent to which the base b is raised in order to obtain a specific value. In other words, y=logbx is equivalent to by=x.

logarithm base b | The exponent to which the base b is raised in order to obtain a specific value. In other words, y=logbx is equivalent to by=x.

logarithmic equation | An equation that involves a logarithm with a variable argument.

logarithmic equation | An equation that involves a logarithm with a variable argument.

mathematical modeling | Using data to find mathematical equations that describe, or model, real-world applications.

mathematical modeling | Using data to find mathematical equations that describe, or model, real-world applications.

midpoint | Given two points (x1,y1) and (x2,y2), the midpoint is an ordered pair given by (x1+x22,y1+y22).

midpoint | Given two points (x1,y1) and (x2,y2), the midpoint is an ordered pair given by (x1+x22,y1+y22).

minor | The determinant of the matrix that results after eliminating a row and column of a square matrix.

minor | The determinant of the matrix that results after eliminating a row and column of a square matrix.

monomial | Polynomial with one term.

natural logarithm | The logarithm base e, denoted lnx.

natural logarithm | The logarithm base e, denoted lnx.

negative exponents | xn=1xn, given any integer n, where x is nonzero.

negative exponents | xn=1xn, given any integer n, where x is nonzero.

negative infinity | The symbol −∞ indicates the interval is unbounded to the left.

negative infinity | The symbol −∞ indicates the interval is unbounded to the left.

non-rigid transformation | A set of operations that change the size and/or shape of a graph in a coordinate plane.

non-rigid transformation | A set of operations that change the size and/or shape of a graph in a coordinate plane.

nonlinear system | A system of equations where at least one equation is not linear.

nonlinear system | A system of equations where at least one equation is not linear.

nth partial sum of a geometric sequence | The sum of the first n terms of a geometric sequence, given by the formula: Sn=a1(1rn)1r, r1.

nth partial sum of a geometric sequence | The sum of the first n terms of a geometric sequence, given by the formula: Sn=a1(1rn)1r, r1.

nth partial sum of an arithmetic sequence | The sum of the first n terms of an arithmetic sequence given by the formula: Sn=n(a1+an)2.

nth partial sum of an arithmetic sequence | The sum of the first n terms of an arithmetic sequence given by the formula: Sn=n(a1+an)2.

nth root | A number that when raised to the nth power (n2) yields the original number.

nth root | A number that when raised to the nth power (n2) yields the original number.

odd integers | Nonzero integers that are not divisible by 2.

odd integers | Nonzero integers that are not divisible by 2.

one-to-one property of exponential functions | Given b>0 and b1 we have bx=by if and only if x=y.

one-to-one property of exponential functions | Given b>0 and b1 we have bx=by if and only if x=y.

one-to-one property of logarithms | Given b>0 and b1 where x,y>0 we have logbx=logby if and only if x=y.

one-to-one property of logarithms | Given b>0 and b1 where x,y>0 we have logbx=logby if and only if x=y.

opposite | Real numbers whose graphs are on opposite sides of the origin with the same distance to the origin.

opposite | Real numbers whose graphs are on opposite sides of the origin with the same distance to the origin.

opposite binomial property | If given a binomial ab, then the opposite is (ab)=ba.

opposite binomial property | If given a binomial ab, then the opposite is (ab)=ba.

opposite reciprocals | Two real numbers whose product is −1. Given a real number ab, the opposite reciprocal is ba.

opposite reciprocals | Two real numbers whose product is −1. Given a real number ab, the opposite reciprocal is ba.

ordered triple | Triples (x, y, z) that identify position relative to the origin in three-dimensional space.

ordered triple | Triples (x, y, z) that identify position relative to the origin in three-dimensional space.

origin | The point on the number line that represents zero.

origin | The point on the number line that represents zero.

origin | The point where the x- and y-axes cross, denoted by (0, 0).

origin | The point where the x- and y-axes cross, denoted by (0, 0).

parabola | The curved graph formed by the squaring function.

parabola | The curved graph formed by the squaring function.

parabola | The U-shaped graph of any quadratic function defined by f(x)=ax2+bx+c, where a, b, and c are real numbers and a0.

parabola | The U-shaped graph of any quadratic function defined by f(x)=ax2+bx+c, where a, b, and c are real numbers and a0.

parabola | The set of points in a plane equidistant from a given line, called the directrix, and a point not on the line, called the focus.

parabola | The set of points in a plane equidistant from a given line, called the directrix, and a point not on the line, called the focus.

parabola in standard form | The equation of a parabola written in the form y=a(xh)2+k or x=a(yk)2+h.

parabola in standard form | The equation of a parabola written in the form y=a(xh)2+k or x=a(yk)2+h.

parallel lines | Lines in the same plane that do not intersect; their slopes are the same.

parallel lines | Lines in the same plane that do not intersect; their slopes are the same.

partial sum | The sum of the first n terms in a sequence denoted Sn.

partial sum | The sum of the first n terms in a sequence denoted Sn.

Pascal’s triangle | A triangular array of numbers that correspond to the binomial coefficients.

Pascal’s triangle | A triangular array of numbers that correspond to the binomial coefficients.

perfect square trinomials | The trinomials obtained by squaring the binomials (a+b)2=a2+2ab+b2 and (ab)2=a22ab+b2.

perfect square trinomials | The trinomials obtained by squaring the binomials (a+b)2=a2+2ab+b2 and (ab)2=a22ab+b2.

piecewise definition | A definition that changes depending on the value of the variable.

piecewise definition | A definition that changes depending on the value of the variable.

placeholders | Terms with zero coefficients used to fill in all missing exponents within a polynomial.

placeholders | Terms with zero coefficients used to fill in all missing exponents within a polynomial.

plane | Any flat two-dimensional surface.

plane | Any flat two-dimensional surface.

plotting points | A way of determining a graph using a finite number of representative ordered pair solutions.

plotting points | A way of determining a graph using a finite number of representative ordered pair solutions.

point-slope form | Any nonvertical line can be written in the form yy1=m(xx1), where m is the slope and (x1,y1) is any point on the line.

point-slope form | Any nonvertical line can be written in the form yy1=m(xx1), where m is the slope and (x1,y1) is any point on the line.

polynomial | An algebraic expression consisting of terms with real number coefficients and variables with whole number exponents.

polynomial | An algebraic expression consisting of terms with real number coefficients and variables with whole number exponents.

polynomial inequality | A mathematical statement that relates a polynomial expression as either less than or greater than another.

polynomial inequality | A mathematical statement that relates a polynomial expression as either less than or greater than another.

polynomial long division | The process of dividing two polynomials using the division algorithm.

polynomial long division | The process of dividing two polynomials using the division algorithm.

polynomials with one variable | A polynomial where each term has the form anxn, where an is any real number and n is any whole number.

polynomials with one variable | A polynomial where each term has the form anxn, where an is any real number and n is any whole number.

power property of equality | Given any positive integer n and real numbers a and b where a=b, then an=bn.

power property of equality | Given any positive integer n and real numbers a and b where a=b, then an=bn.

power property of logarithms | logbxn=nlogbx; the logarithm of a quantity raised to a power is equal to that power times the logarithm of the quantity.

power property of logarithms | logbxn=nlogbx; the logarithm of a quantity raised to a power is equal to that power times the logarithm of the quantity.

power rule for exponents | (xm)n=xmn; a power raised to a power can be simplified by multiplying the exponents.

power rule for exponents | (xm)n=xmn; a power raised to a power can be simplified by multiplying the exponents.

prime factorization | The unique factorization of a natural number written as a product of primes.

prime factorization | The unique factorization of a natural number written as a product of primes.

prime number | Integer greater than 1 that is divisible only by 1 and itself.

prime number | Integer greater than 1 that is divisible only by 1 and itself.

prime polynomial | A polynomial with integer coefficients that cannot be factored as a product of polynomials with integer coefficients other than 1 and itself.

prime polynomial | A polynomial with integer coefficients that cannot be factored as a product of polynomials with integer coefficients other than 1 and itself.

principal (nonnegative) nth root | The positive nth root when n is even.

principal (nonnegative) nth root | The positive nth root when n is even.

principal (nonnegative) square root | The non-negative square root.

principal (nonnegative) square root | The non-negative square root.

principal (nonnegative) square root | The positive square root of a positive real number, denoted with the symbol .

principal (nonnegative) square root | The positive square root of a positive real number, denoted with the symbol .

product of complex conjugates | The real number that results from multiplying complex conjugates: (a+bi)(abi)=a2+b2.

product of complex conjugates | The real number that results from multiplying complex conjugates: (a+bi)(abi)=a2+b2.

product property of logarithms | logb(xy)=logbx+logby; the logarithm of a product is equal to the sum of the logarithm of the factors.

product property of logarithms | logb(xy)=logbx+logby; the logarithm of a product is equal to the sum of the logarithm of the factors.

product rule for exponents | xmxn=xm+n; the product of two expressions with the same base can be simplified by adding the exponents.

product rule for exponents | xmxn=xm+n; the product of two expressions with the same base can be simplified by adding the exponents.

profit function | A function that models the profit as revenue less cost.

profit function | A function that models the profit as revenue less cost.

properties of equality | Properties that allow us to obtain equivalent equations by adding, subtracting, multiplying, and dividing both sides of an equation by nonzero real numbers.

properties of equality | Properties that allow us to obtain equivalent equations by adding, subtracting, multiplying, and dividing both sides of an equation by nonzero real numbers.

properties of inequalities | Properties used to obtain equivalent inequalities and used as a means to solve them.

properties of inequalities | Properties used to obtain equivalent inequalities and used as a means to solve them.

proportion | A statement of equality of two ratios.

proportion | A statement of equality of two ratios.

Pythagorean theorem | The hypotenuse of any right triangle is equal to the square root of the sum of the squares of the lengths of the triangle’s legs.

Pythagorean theorem | The hypotenuse of any right triangle is equal to the square root of the sum of the squares of the lengths of the triangle’s legs.

quadrants | The four regions of a rectangular coordinate plane partly bounded by the x- and y-axes and numbered using the Roman numerals I, II, III, and IV.

quadrants | The four regions of a rectangular coordinate plane partly bounded by the x- and y-axes and numbered using the Roman numerals I, II, III, and IV.

quadratic form | An equation of the form au2+bu+c=0 where a, b and c are real numbers and u represents an algebraic expression.

quadratic form | An equation of the form au2+bu+c=0 where a, b and c are real numbers and u represents an algebraic expression.

quadratic formula | The formula x=b±b24ac2a, which gives the solutions to any quadratic equation in the standard form ax2+bx+c=0, where a, b, and c are real numbers and a0.

quadratic formula | The formula x=b±b24ac2a, which gives the solutions to any quadratic equation in the standard form ax2+bx+c=0, where a, b, and c are real numbers and a0.

quadratic inequality | A mathematical statement that relates a quadratic expression as either less than or greater than another.

quadratic inequality | A mathematical statement that relates a quadratic expression as either less than or greater than another.

quotient | The result of division.

quotient | The result of dividing.

quotient | The result of division.

quotient | The result of dividing.

quotient property of logarithms | logb(xy)=logbxlogby; the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.

quotient property of logarithms | logb(xy)=logbxlogby; the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.

quotient rule for radicals | Given real numbers An and Bn, ABn=AnBn where B0.

quotient rule for radicals | Given real numbers An and Bn, ABn=AnBn where B0.

quotients with negative exponents | xnym=ymxn, given any integers m and n, where x0 and y0.

quotients with negative exponents | xnym=ymxn, given any integers m and n, where x0 and y0.

radical | Used when referring to an expression of the form An.

radical | Used when referring to an expression of the form An.

radical equation | Any equation that contains one or more radicals with a variable in the radicand.

radical equation | Any equation that contains one or more radicals with a variable in the radicand.

radical expression | An algebraic expression that contains radicals.

radical expression | An algebraic expression that contains radicals.

radical is simplified | A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.

radical is simplified | A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.

radicand | The number within a radical.

radicand | The number within a radical.

radicand | The expression A within a radical sign, An.

radicand | The expression A within a radical sign, An.

rational equation | An equation containing at least one rational expression.

rational equation | An equation containing at least one rational expression.

rational inequality | A mathematical statement that relates a rational expression as either less than or greater than another.

rational inequality | A mathematical statement that relates a rational expression as either less than or greater than another.

rationalizing the denominator | The process of determining an equivalent radical expression with a rational denominator.

rationalizing the denominator | The process of determining an equivalent radical expression with a rational denominator.

real numbers | The set of all rational and irrational numbers.

real numbers | The set of all rational and irrational numbers.

reciprocal function | The function defined by f(x)=1x.

reciprocal function | The function defined by f(x)=1x.

reciprocals | Two real numbers whose product is 1.

reciprocals | Two real numbers whose product is 1.

recurrence relation | A formula that uses previous terms of a sequence to describe subsequent terms.

recurrence relation | A formula that uses previous terms of a sequence to describe subsequent terms.

reducing | The process of finding equivalent fractions by dividing the numerator and the denominator by common factors.

reducing | The process of finding equivalent fractions by dividing the numerator and the denominator by common factors.

reflection | A transformation that produces a mirror image of the graph about an axis.

reflection | A transformation that produces a mirror image of the graph about an axis.

relation | Any set of ordered pairs.

relation | Any set of ordered pairs.

relatively prime | Expressions that share no common factors other than 1.

relatively prime | Expressions that share no common factors other than 1.

Restrictions | The set of real numbers for which a rational function is not defined.

Restrictions | The set of real numbers for which a rational function is not defined.

revenue function | A function that models income based on a number of units sold.

revenue function | A function that models income based on a number of units sold.

root | A value in the domain of a function that results in zero.

root | A value in the domain of a function that results in zero.

row echelon form | A matrix in triangular form where the leading nonzero element of each row is 1.

row echelon form | A matrix in triangular form where the leading nonzero element of each row is 1.

run | The horizontal change between any two points on a line.

run | The horizontal change between any two points on a line.

scientific notation | Real numbers expressed the form a×10n, where n is an integer and 1a<10.

scientific notation | Real numbers expressed the form a×10n, where n is an integer and 1a<10.

secant line | Line that intersects two points on the graph of a function.

secant line | Line that intersects two points on the graph of a function.

set notation | Notation used to describe a set using mathematical symbols.

set notation | Notation used to describe a set using mathematical symbols.

sign chart | A model of a function using a number line and signs (+ or −) to indicate regions in the domain where the function is positive or negative.

sign chart | A model of a function using a number line and signs (+ or −) to indicate regions in the domain where the function is positive or negative.

similar radicals | Term used when referring to like radicals.

similar radicals | Term used when referring to like radicals.

similar terms | Used when referring to like terms.

similar terms | Used when referring to like terms.

Simple interest | Modeled by the formula I=prt, where p represents the principal amount invested at an annual interest rate r for t years.

Simple interest | Modeled by the formula I=prt, where p represents the principal amount invested at an annual interest rate r for t years.

simplified radical | A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.

simplified radical | A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.

simplifying the expression | The process of combining like terms until the expression contains no more similar terms.

simplifying the expression | The process of combining like terms until the expression contains no more similar terms.

simultaneous solution | Used when referring to a solution of a system of equations.

simultaneous solution | Used when referring to a solution of a system of equations.

slope formula | The slope of the line through the points (x1,y1) and (x2,y2) is given by the formula m=y2y1x2x1.

slope formula | The slope of the line through the points (x1,y1) and (x2,y2) is given by the formula m=y2y1x2x1.

slope-intercept form | Any nonvertical line can be written in the form y=mx+b, where m is the slope and (0, b) is the y-intercept.

slope-intercept form | Any nonvertical line can be written in the form y=mx+b, where m is the slope and (0, b) is the y-intercept.

solution | Any value that can replace the variable in an equation to produce a true statement.

solution | Any value that can replace the variable in an equation to produce a true statement.

solution to a linear inequality | A real number that produces a true statement when its value is substituted for the variable.

solution to a linear inequality | A real number that produces a true statement when its value is substituted for the variable.

Solutions | Values that can be used in place of the variable to satisfy the given condition.

Solutions | Values that can be used in place of the variable to satisfy the given condition.

solve by factoring | The process of solving an equation that is equal to zero by factoring it and then setting each variable factor equal to zero.

solve by factoring | The process of solving an equation that is equal to zero by factoring it and then setting each variable factor equal to zero.

split function | A term used when referring to a piecewise function.

split function | A term used when referring to a piecewise function.

square matrix | A matrix with the same number of rows and columns.

square matrix | A matrix with the same number of rows and columns.

square root function | The function defined by f(x)=x.

square root function | The function defined by f(x)=x.

square root function | The function defined by f(x)=x.

square root function | The function defined by f(x)=x.

square root property | For any real number k, if x2=k, then x=±k.

square root property | For any real number k, if x2=k, then x=±k.

squaring function | The quadratic function defined by f(x)=x2.

squaring function | The quadratic function defined by f(x)=x2.

squaring property of equality | Given real numbers a and b, where a=b, then a2=b2.

squaring property of equality | Given real numbers a and b, where a=b, then a2=b2.

standard form | Any quadratic equation in the form ax2+bx+c=0, where a, b, and c are real numbers and a0.

standard form | Any quadratic equation in the form ax2+bx+c=0, where a, b, and c are real numbers and a0.

substitute | The act of replacing a variable with an equivalent quantity.

substitute | The act of replacing a variable with an equivalent quantity.

substitution method | A means of solving a linear system by solving for one of the variables and substituting the result into the other equation.

substitution method | A means of solving a linear system by solving for one of the variables and substituting the result into the other equation.

subtraction | Subtract functions as indicated by the notation: (fg)(x)=f(x)g(x).

subtraction | Subtract functions as indicated by the notation: (fg)(x)=f(x)g(x).

sum of squares | a2+b2, where a and b represent algebraic expressions. This does not have a general factored equivalent.

sum of squares | a2+b2, where a and b represent algebraic expressions. This does not have a general factored equivalent.

summation | Used when referring to sigma notation.

summation | Used when referring to sigma notation.

symmetric property | Allows you to solve for the variable on either side of the equal sign, because x=5 is equivalent to 5=x.

symmetric property | Allows you to solve for the variable on either side of the equal sign, because x=5 is equivalent to 5=x.

system of inequalities | A set of two or more inequalities with the same variables.

system of inequalities | A set of two or more inequalities with the same variables.

TBA | Given any real number a, a+0=0+a=a.

TBA | Given any real number a, a+(a)=(a)+a=0.

TBA | Given real numbers a, b and c, (a+b)+c=a+(b+c).

TBA | Given real numbers a and b, a+b=b+a.

TBA | Given any real number a, a0=0a=0.

TBA | Given any real number a, a1=1a=a.

TBA | Given any real numbers a, b and c, (ab)c=a(bc).

TBA | Given any real numbers a and b, ab=ba.

TBA | Given real numbers An and Bn, ABn=AnBn.

TBA | Given real numbers An and Bn, ABn=AnBn.

TBA | (xy)n=xnyn; if a product is raised to a power, then apply that power to each factor in the product.

TBA | (xy)n=xnyn; if a quotient is raised to a power, then apply that power to the numerator and the denominator.

TBA | xmxn=xmn; the quotient of two expressions with the same base can be simplified by subtracting the exponents.

TBA | A polynomial with degree 1.

TBA | A polynomial with degree 2.

TBA | A polynomial with degree 3.

TBA | Given any real number a, a+0=0+a=a.

TBA | Given any real number a, a+(a)=(a)+a=0.

TBA | Given real numbers a, b and c, (a+b)+c=a+(b+c).

TBA | Given real numbers a and b, a+b=b+a.

TBA | Given any real number a, a0=0a=0.

TBA | Given any real number a, a1=1a=a.

TBA | Given any real numbers a, b and c, (ab)c=a(bc).

TBA | Given any real numbers a and b, ab=ba.

TBA | Given real numbers An and Bn, ABn=AnBn.

TBA | Given real numbers An and Bn, ABn=AnBn.

TBA | (xy)n=xnyn; if a product is raised to a power, then apply that power to each factor in the product.

TBA | (xy)n=xnyn; if a quotient is raised to a power, then apply that power to the numerator and the denominator.

TBA | xmxn=xmn; the quotient of two expressions with the same base can be simplified by subtracting the exponents.

TBA | Used when referring to direct variation.

TBA | Used when referring to joint variation.

TBA | Used when referring to direct variation.

TBA | Used when referring to joint variation.

test points | A point not on the boundary of the linear inequality used as a means to determine in which half-plane the solutions lie.

test points | A point not on the boundary of the linear inequality used as a means to determine in which half-plane the solutions lie.

trial and error (or guess and check) method | Describes the method of factoring a trinomial by systematically checking factors to see if their product is the original trinomial.

trial and error (or guess and check) method | Describes the method of factoring a trinomial by systematically checking factors to see if their product is the original trinomial.

trinomial | Polynomial with three terms.

u-substitution | A technique in algebra using substitution to transform equations into familiar forms.

u-substitution | A technique in algebra using substitution to transform equations into familiar forms.

undefined | A quotient such as 50 is left without meaning and is not assigned an interpretation.

undefined | A quotient such as 50 is left without meaning and is not assigned an interpretation.

uniform motion | The distance D after traveling at an average rate r for some time t can be calculated using the formula D=rt.

uniform motion | The distance D after traveling at an average rate r for some time t can be calculated using the formula D=rt.

Uniform motion (or distance) | Described by the formula D=rt, where the distance D is given as the product of the average rate r and the time t traveled at that rate.

Uniform motion (or distance) | Described by the formula D=rt, where the distance D is given as the product of the average rate r and the time t traveled at that rate.

union | The set formed by joining the individual solution sets indicated by the logical use of the word “or” and denoted with the symbol .

union | The set formed by joining the individual solution sets indicated by the logical use of the word “or” and denoted with the symbol .

unit circle | The circle centered at the origin with radius 1; its equation is x2+y2=1.

unit circle | The circle centered at the origin with radius 1; its equation is x2+y2=1.

upper triangular form | A linear system consisting of equations with three variables in standard form arranged so that the variable x does not appear after the first equation and the variable y does not appear after the second equation.

upper triangular form | A linear system consisting of equations with three variables in standard form arranged so that the variable x does not appear after the first equation and the variable y does not appear after the second equation.

variables | Letters used to represent numbers.

variables | Letters used to represent numbers.

vertex form | A quadratic function written in the form f(x)=a(xh)2+k.

vertex form | A quadratic function written in the form f(x)=a(xh)2+k.

vertex form | The equation of a parabola written in standard form is often called vertex form. In this form the vertex is apparent: (h,k).

vertex form | The equation of a parabola written in standard form is often called vertex form. In this form the vertex is apparent: (h,k).

vertical asymptote | A vertical line to which a graph becomes infinitely close.

vertical asymptote | A vertical line to which a graph becomes infinitely close.

vertical line test | If any vertical line intersects the graph more than once, then the graph does not represent a function.

vertical line test | If any vertical line intersects the graph more than once, then the graph does not represent a function.

vertical translation | A rigid transformation that shifts a graph up or down.

vertical translation | A rigid transformation that shifts a graph up or down.

vertices. | Points on the separate branches of a hyperbola where the distance is a minimum.

vertices. | Points on the separate branches of a hyperbola where the distance is a minimum.

whole numbers | The set of natural numbers combined with zero: {0, 1, 2, 3, 4, 5, …}.

whole numbers | The set of natural numbers combined with zero: {0, 1, 2, 3, 4, 5, …}.

work rate | The rate at which a task can be performed.

work rate | The rate at which a task can be performed.

work-rate formula | 1t1t+1t2t=1, where 1t1 and 1t2 are the individual work rates and t is the time it takes to complete the task working together.

work-rate formula | 1t1t+1t2t=1, where 1t1 and 1t2 are the individual work rates and t is the time it takes to complete the task working together.

x-intercept | The point (or points) where a graph intersects the x-axis, expressed as an ordered pair (x, 0).

x-intercept | The point (or points) where a graph intersects the x-axis, expressed as an ordered pair (x, 0).

y-intercept | The point (or points) where a graph intersects the y-axis, expressed as an ordered pair (0, y).

y-intercept | The point (or points) where a graph intersects the y-axis, expressed as an ordered pair (0, y).

zero as an exponent | x0=1; any nonzero base raised to the 0 power is defined to be 1.

zero as an exponent | x0=1; any nonzero base raised to the 0 power is defined to be 1.

zero factorial | The factorial of zero is defined to be equal to 1; 0!=1.

zero factorial | The factorial of zero is defined to be equal to 1; 0!=1.

zero-product property | A product is equal to zero if and only if at least one of the factors is zero.

zero-product property | A product is equal to zero if and only if at least one of the factors is zero.

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