Glossary

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Example and Directions
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Glossary Entries
Word(s)	Definition	Image	Caption	Link	Source
absolute value	The absolute value of a number is the distance from the graph of the number to zero on a number line.
absolute value	The absolute value of a number is the distance from the graph of the number to zero on a number line.
AC method	Method for factoring trinomials by replacing the middle term with two terms that allow us to factor the resulting four-term polynomial by grouping.
AC method	Method for factoring trinomials by replacing the middle term with two terms that allow us to factor the resulting four-term polynomial by grouping.
add polynomials	The process of combining all like terms of two or more polynomials.
add polynomials	The process of combining all like terms of two or more polynomials.
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.
Additive identity property	Given any real number a, $a + 0 = 0 + a = a .$
Additive identity property	Given any real number a, $a + 0 = 0 + a = a .$
Additive inverse property	Given any real number a, $a + (- a) = (- a) + a = 0 .$
Additive inverse property	Given any real number a, $a + (- a) = (- a) + a = 0 .$
algebraic expressions	Combinations of variables and numbers along with mathematical operations used to generalize specific arithmetic operations.
algebraic expressions	Combinations of variables and numbers along with mathematical operations used to generalize specific arithmetic operations.
algebraic fraction	Term used when referring to a rational expression.
algebraic fraction	Term used when referring to a rational expression.
Area of a circle	$A = π r^{2}$ , where r represents the radius and the constant $π \approx 3.14$ .
Area of a circle	$A = π r^{2}$ , where r represents the radius and the constant $π \approx 3.14$ .
Area of a rectangle	$A = l w$ , where l represents the length and w represents the width.
Area of a rectangle	$A = l w$ , where l represents the length and w represents the width.
Area of a square	$A = s^{2}$ , where s represents the length of each side.
Area of a square	$A = s^{2}$ , where s represents the length of each side.
Area of a triangle	$A = \frac{1}{2} b h$ , where b represents the length of the base and h represents the height.
Area of a triangle	$A = \frac{1}{2} b h$ , where b represents the length of the base and h represents the height.
Associative property	Given real numbers a, b and c, $(a + b) + c = a + (b + c)$ .
Associative property	Given any real numbers a, b, and c, $(a \cdot b) \cdot c = a \cdot (b \cdot c) .$
Associative property	Given real numbers a, b and c, $(a + b) + c = a + (b + c)$ .
Associative property	Given any real numbers a, b, and c, $(a \cdot b) \cdot c = a \cdot (b \cdot c) .$
asterisk	The symbol (*) that indicates multiplication within text-based applications.
asterisk	The symbol (*) that indicates multiplication within text-based applications.
average	Used in reference to the arithmetic mean.
average	Used in reference to the arithmetic mean.
average cost	The total cost divided by the number of units produced, which can be represented by $c (x) = \frac{C (x)}{x}$ , where $C (x)$ is a cost function.
average cost	The total cost divided by the number of units produced, which can be represented by $c (x) = \frac{C (x)}{x}$ , 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 their equivalent equations, 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 their equivalent equations, to determine the corresponding value of the other variable.
back substituting	The process of finding the answers to other unknowns after one has been found.
back substituting	The process of finding the answers to other unknowns after one has been found.
Binomial	Polynomial with two terms.
Binomial	Polynomial with two terms.
caret	The symbol ^ that indicates exponents on many calculators, $a^{n} = a^n$ .
caret	The symbol ^ that indicates exponents on many calculators, $a^{n} = a^n$ .
Cartesian coordinate system	Used in honor of René Descartes when referring to the rectangular coordinate system.
Cartesian coordinate system	Used in honor of René Descartes when referring to the rectangular coordinate system.
check by evaluating	We can be fairly certain that we have multiplied the polynomials correctly if we check that a few values evaluate to the same results in the original expression and in the answer.
check by evaluating	We can be fairly certain that we have multiplied the polynomials correctly if we check that a few values evaluate to the same results in the original expression and in the answer.
circumference	The perimeter of a circle given by $C = 2 π r$ , where r represents the radius of the circle and $π \approx 3.14159$ .
circumference	The perimeter of a circle given by $C = 2 π r$ , where r represents the radius of the circle and $π \approx 3.14159$ .
collinear	Describes points that lie on the same line.
collinear	Describes points that lie on the same line.
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 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.
Commutative property	Given real numbers a and b, $a + b = b + a$ .
Commutative property	Given any real numbers a and b, $a \cdot b = b \cdot a .$
Commutative property	Given real numbers a and b, $a + b = b + a$ .
Commutative property	Given any real numbers a and b, $a \cdot b = b \cdot a .$
completing the square	The process of rewriting a quadratic equation in the form ${(x - p)}^{2} = q$ .
completing the square	The process of rewriting a quadratic equation in the form ${(x - p)}^{2} = q$ .
complex conjugate	Two complex numbers whose real parts are the same and imaginary parts are opposite. If given $a + b i$ , then its complex conjugate is $a - b i$ .
complex conjugate	Two complex numbers whose real parts are the same and imaginary parts are opposite. If given $a + b i$ , then its complex conjugate is $a - b i$ .
complex fraction	A fraction where the numerator or denominator consists of one or more fractions.
complex fraction	A fraction where the numerator or denominator consists of one or more fractions.
complex fraction	A fraction where the numerator or denominator consists of one or more fractions.
complex fraction	A fraction where the numerator or denominator consists of one or more fractions.
complex number	Numbers of the form $a + b i$ , where a and b are real numbers.
complex number	Numbers of the form $a + b i$ , where a and b are real numbers.
complex rational expression	A rational expression where the numerator or denominator consists of one or more rational expressions.
complex rational expression	A rational expression where the numerator or denominator consists of one or more rational expressions.
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.”
conjugate binomials	The binomials $(a + b)$ and $(a - b)$ .
conjugate binomials	The binomials $(a + b)$ and $(a - b)$ .
conjugates	The factors $(a + b)$ and $(a - b)$ are conjugates.
conjugates	The factors $(a + b)$ and $(a - b)$ are conjugates.
Constant function	A polynomial function with degree 0.
Constant function	A polynomial function with degree 0.
constant of proportionality	Used when referring to the constant of variation.
constant of proportionality	Used when referring to the constant of variation.
constant term	A term written without a variable factor.
constant term	A term written without a variable factor.
contradiction	An equation that is never true and has no solution.
contradiction	An equation that is never true and has no solution.
cross canceling	Cancelling common factors in the numerator and the denominator of fractions before multiplying.
cross canceling	Cancelling common factors in the numerator and the denominator of fractions before multiplying.
cross multiplication	If $\frac{a}{b} = \frac{c}{d}$ , then $a d = b c$ .
cross multiplication	If $\frac{a}{b} = \frac{c}{d}$ , then $a d = b c$ .
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 $f (x) = \sqrt[3]{x}$ .
cube root function	The function $f (x) = \sqrt[3]{x}$ .
Cubic function	A polynomial function with degree 3.
Cubic function	A polynomial function with degree 3.
decimal	A real number expressed using the decimal system.
decimal	A real number expressed using the decimal system.
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 system that consists of equivalent equations with infinitely many ordered pair solutions, denoted by (x, mx + b).
dependent system	A system that consists of equivalent equations with 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 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 as the dependent variable.
difference of cubes	$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}),$ where a and b represent algebraic expressions.
difference of cubes	$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}),$ where a and b represent algebraic expressions.
difference of squares	$a^{2} - b^{2} = (a + b) (a - b),$ where a and b represent algebraic expressions.
difference of squares	$a^{2} - b^{2} = (a + b) (a - b),$ where a and b represent algebraic expressions.
difference of squares	$a^{2} - b^{2} = (a + b) (a - b),$ where a and b represent algebraic expressions.
difference of squares	$a^{2} - b^{2} = (a + b) (a - b),$ where a and b represent algebraic expressions.
direct variation	Describes two quantities x and y that are constant multiples of each other: $y = k x$ .
direct variation	Describes two quantities x and y that are constant multiples of each other: $y = k x$ .
directly proportional	Used when referring to direct variation.
directly proportional	Used when referring to direct variation.
discriminant	The algebraic expression $b^{2} - 4 a c$ .
discriminant	The algebraic expression $b^{2} - 4 a c$ .
discriminant	The expression inside the radical of the quadratic formula, $b^{2} - 4 a c$ .
discriminant	The expression inside the radical of the quadratic formula, $b^{2} - 4 a c$ .
distance formula	Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , calculate the distance d between them using the formula $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$
distance formula	Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , calculate the distance d between them using the formula $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$
Distance formula	Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2}),$ calculate the distance d between them using the formula d = $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$
Distance formula	Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2}),$ calculate the distance d between them using the formula d = $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$
distance formula for a number line	The distance between any two real numbers a and b on a number line can be calculated using the formula $d = \| b - a \|$ .
distance formula for a number line	The distance between any two real numbers a and b on a number line can be calculated using the formula $d = \| b - a \|$ .
distributive property	Given any real numbers a, b, and c, $a (b + c) = a b + a c$ or $(b + c) a = b a + c a$ .
distributive property	Given any real numbers a, b, and c, $a (b + c) = a b + a c$ or $(b + c) a = b a + c a$ .
dividend	The numerator of a quotient.
dividend	The numerator of a quotient.
divisor	The denominator of a quotient.
divisor	The denominator of a quotient.
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.
elimination (or addition) method	A means of solving a system by adding equivalent equations in such a way as to eliminate a variable.
elimination (or addition) method	A means of solving a system by adding equivalent equations in such a way as to eliminate a variable.
empty set	A subset with no elements, denoted $\emptyset$ or { }.
empty set	A subset with no elements, denoted $\emptyset$ or { }.
equality relationship	Express equality with the symbol =. If two quantities are not equal, use the symbol $\neq$ .
equality relationship	Express equality with the symbol =. If two quantities are not equal, use the symbol $\neq$ .
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 two or are multiples of two.
even integers	Integers that are divisible by two or are multiples of two.
exponent	The positive integer n in the exponential notation $a^{n}$ that indicates the number of times the base is used as a factor.
exponent	The positive integer n in the exponential notation $a^{n}$ 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 form	An equivalent expression written using a rational exponent.
exponential form	An equivalent expression written using a rational exponent.
exponential notation	The compact notation $a x^{2} + b x + c = 0.$ used when a factor is repeated multiple times.
exponential notation	The compact notation $a x^{2} + b x + c = 0.$ used when a factor is repeated multiple times.
extracting the roots	Applying the square root property as a means of solving a quadratic equation.
extracting the roots	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 solution that does not solve the original equation.
extraneous solutions	A solution that does not solve the original equation.
factor by grouping	A technique for factoring polynomials with four terms.
factor by grouping	A technique for factoring polynomials with four terms.
factoring a polynomial	The process of rewriting a polynomial as a product of polynomial factors.
factoring a polynomial	The process of rewriting a polynomial as a product of polynomial factors.
Factoring out the GCF	The process of rewriting a polynomial as a product using the GCF of all of its terms.
Factoring out the GCF	The process of rewriting a polynomial as a product using the GCF of all of its terms.
factors	Any of the numbers or expressions 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.
factors	Any of the numbers or expressions that form a product.
FOIL	When multiplying binomials we apply the distributive property multiple times in such a way as to multiply the first terms, outer terms, inner terms, and last terms.
FOIL	When multiplying binomials we apply the distributive property multiple times in such a way as to multiply the first terms, outer terms, inner terms, and last terms.
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.
functions	Relations where every x-value corresponds to exactly one y-value. With the definition comes new notation: $f (x) = y$ , which is read “f of x is equal to y.”
functions	Relations where every x-value corresponds to exactly one y-value. With the definition comes new notation: $f (x) = y$ , which is read “f of x is equal to y.”
fundamental theorem of algebra	Guarantees that there will be as many (or fewer) real solutions to a polynomial with one variable as its degree.
fundamental theorem of algebra	Guarantees that there will be as many (or fewer) real solutions to a polynomial with one variable as its degree.
GCF of a polynomial	The greatest common factor of all the terms of the polynomial.
GCF of a polynomial	The greatest common factor of all the terms of the polynomial.
GCF of monomials	The product of the GCF of the coefficients and all common variable factors.
GCF of monomials	The product of the GCF of the coefficients and all common variable factors.
graph	A point on the number line associated with a coordinate.
graph	A point on the number line associated with a coordinate.
graph	A point on the number line associated with a coordinate.
graph	A point on the number line associated with a coordinate.
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 factor (GCF)	The product of all the common prime factors.
greatest common factor (GCF)	The product of all the common prime factors.
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.
guess and check	Used when referring to the trial and error method for factoring trinomials.
guess and check	Used when referring to the trial and error method for factoring trinomials.
horizontal line	Any line whose equation can be written in the form y = k, where k is a real number.
horizontal line	Any line whose equation can be written in the form y = k, where k is a real number.
identity	An equation that is true for all possible values.
identity	An equation that is true for all possible values.
imaginary numbers	The square roots of any negative real numbers.
imaginary numbers	The square roots of any negative real numbers.
imaginary part	The real number b of a complex number $a + b i$ .
imaginary part	The real number b of a complex number $a + b i$ .
imaginary unit	Defined as $i = \sqrt{- 1}$ and $i^{2} = - 1$ .
imaginary unit	Defined as $i = \sqrt{- 1}$ and $i^{2} = - 1$ .
inclusive inequalities	Use the symbol $\leq$ to express quantities that are “less than or equal to” and $\geq$ for quantities that are “greater than or equal to” each other.
inclusive inequalities	Use the symbol $\leq$ to express quantities that are “less than or equal to” and $\geq$ for quantities that are “greater than or equal to” each other.
inconsistent systems	A system with no simultaneous solution.
inconsistent systems	A system with no simultaneous solution.
independent systems	A system of equations with one ordered pair solution (x, y).
independent systems	A system of equations with one ordered pair solution (x, y).
indeterminate	A quotient such as $\frac{0}{0}$ , which is a quantity that is uncertain or ambiguous.
indeterminate	A quotient such as $\frac{0}{0}$ , which is a quantity that is uncertain or ambiguous.
index	The positive integer n in the notation $\sqrt[n]{}$ that is used to indicate an nth root.
index	The positive integer n in the notation $\sqrt[n]{}$ that is used to indicate an nth root.
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, …}.
interest and money problems	Applications involving simple interest and money.
interest and money problems	Applications involving simple interest and money.
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 $\cap$ .
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 $\cap$ .
interval notation	A textual system of expressing solutions to an algebraic inequality.
interval notation	A textual system of expressing solutions to an algebraic inequality.
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.
jointly proportional	Used when referring to joint variation.
jointly proportional	Used when referring to joint variation.
leading coefficient	The coefficient of the term with the largest degree.
leading coefficient	The coefficient of the term with the largest degree.
least common denominator (LCD)	The least common multiple of a set of denominators.
least common denominator (LCD)	The least common multiple of a set of denominators.
least common multiple (LCM)	The smallest number that is evenly divisible by a set of numbers.
least common multiple (LCM)	The smallest number that is evenly divisible by a set of numbers.
line graph	A set of related data values graphed on a coordinate plane and connected by line segments.
line graph	A set of related data values graphed on a coordinate plane and connected by line segments.
linear equation with one variable	An equation that can be written in the general form $a x + b = 0$ , where a and b are real numbers and $a \neq 0$ .
linear equation with one variable	An equation that can be written in the general form $a x + b = 0$ , where a and b are real numbers and $a \neq 0$ .
linear equation with two variables	An equation with two variables that can be written in the standard form $a x + b y = c$ , where the real numbers a and b are not both zero.
linear equation with two variables	An equation with two variables that can be written in the standard form $a x + b y = c$ , where the real numbers a and b are not both zero.
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 function	A polynomial function with degree 1.
Linear function	A polynomial function with degree 1.
linear inequality	A mathematical statement relating a linear expression as either less than or greater than another.
linear inequality	A mathematical statement relating a linear expression as either less than or greater than another.
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	In this section, we restrict our study to systems of two linear equations with two variables.
linear systems	In this section, we restrict our study to systems of two linear equations with two variables.
literal equations	A formula that summarizes whole classes of problems.
literal equations	A formula that summarizes whole classes of problems.
midpoint	Given two points, $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the midpoint is an ordered pair given by $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ .
midpoint	Given two points, $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the midpoint is an ordered pair given by $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ .
mixed number	A number that represents the sum of a whole number and a fraction.
mixed number	A number that represents the sum of a whole number and a fraction.
Mixture problems	Applications involving a mixture of amounts usually given as a percentage of some total.
Mixture problems	Applications involving a mixture of amounts usually given as a percentage of some total.
Monomial	Polynomial with one term.
Monomial	Polynomial with one term.
Multiplicative identity property	Given any real number a, $a \cdot 1 = 1 \cdot a = a .$
Multiplicative identity property	Given any real number a, $a \cdot 1 = 1 \cdot a = a .$
natural (or counting) numbers	The set of counting numbers {1, 2, 3, 4, 5, …}.
natural (or counting) numbers	The set of counting numbers {1, 2, 3, 4, 5, …}.
negative exponents	$x^{- n} = \frac{1}{x^{n}},$ given any integer n, where x is nonzero.
negative exponents	$x^{- n} = \frac{1}{x^{n}},$ 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.
nth root	The number that, when raised to the nth power, yields the original number.
nth root	The number that, when raised to the nth power, yields the original number.
odd integers	Integers that are not divisible by 2.
odd integers	Integers that are not divisible by 2.
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 $a - b$ , then the opposite is $- (a - b) = b - a$ .
opposite binomial property	If given a binomial $a - b$ , then the opposite is $- (a - b) = b - a$ .
opposite reciprocals	Two real numbers whose product is −1. Given a real number $\frac{a}{b}$ , the opposite reciprocal is $- \frac{b}{a}$ .
opposite reciprocals	Two real numbers whose product is −1. Given a real number $\frac{a}{b}$ , the opposite reciprocal is $- \frac{b}{a}$ .
opposite-side like terms	Like terms of an equation on opposite sides of the equal sign.
opposite-side like terms	Like terms of an equation on opposite sides of the equal sign.
order	To ensure a single correct result, perform mathematical operations in a specific order.
order	To ensure a single correct result, perform mathematical operations in a specific order.
origin	The point on the number line that represtents zero.
origin	The point on the number line that represtents 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 graph of any quadratic equation $y = a x^{2} + b x + c$ , where a, b, and c are real numbers and $a \neq 0$ .
parabola	The graph of any quadratic equation $y = a x^{2} + b x + c$ , where a, b, and c are real numbers and $a \neq 0$ .
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.
percent	A representation of a number as part of 100: $N % = \frac{N}{100}$ .
percent	A representation of a number as part of 100: $N % = \frac{N}{100}$ .
perfect cube	The result of cubing an integer.
perfect cube	The result of cubing an integer.
perfect square	The result of squaring an integer.
perfect square	The result of squaring an integer.
perfect square trinomials	The trinomials obtained by squaring the binomials ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ and ${(a - b)}^{2} = a^{2} - 2 a b + b^{2} .$
perfect square trinomials	The trinomials obtained by squaring the binomials ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ and ${(a - b)}^{2} = a^{2} - 2 a b + b^{2} .$
perimeter	The sum of the lengths of all the outside edges of a polygon.
perimeter	The sum of the lengths of all the outside edges of a polygon.
Perimeter of a rectangle	$P = 2 l + 2 w$ , where l represents the length and w represents the width.
Perimeter of a rectangle	$P = 2 l + 2 w$ , where l represents the length and w represents the width.
Perimeter of a square	$P = 4 s$ , where s represents the length of a side.
Perimeter of a square	$P = 4 s$ , where s represents the length of a side.
Perimeter of a triangle	$P = a + b + c$ , where a, b, and c each represents the length of a different side.
Perimeter of a triangle	$P = a + b + c$ , where a, b, and c each represents the length of a different side.
pie chart	A circular graph divided into sectors whose area is proportional to the relative size of the ratio of the part to the total.
pie chart	A circular graph divided into sectors whose area is proportional to the relative size of the ratio of the part to the total.
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.
point-slope form of a line	Any nonvertical line can be written in the form $y - y_{1} = m (x - x_{1})$ , where m is the slope and $(x_{1}, y_{1})$ is any point on the line.
point-slope form of a line	Any nonvertical line can be written in the form $y - y_{1} = m (x - x_{1})$ , where m is the slope and $(x_{1}, y_{1})$ 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	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 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 $a_{n} x^{n}$ , where $a_{n}$ is any real number and n is any whole number.
polynomials with one variable	A polynomial where each term has the form $a_{n} x^{n}$ , where $a_{n}$ 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 $a^{n} = b^{n}$ .
power property of equality	Given any positive integer n and real numbers a and b, where $a = b$ , then $a^{n} = b^{n}$ .
power rule for a product	${(x y)}^{n} = x^{n} y^{n}$ ; if a product is raised to a power, then apply that power to each factor in the product.
power rule for a product	${(x y)}^{n} = x^{n} y^{n}$ ; if a product is raised to a power, then apply that power to each factor in the product.
power rule for a quotient	${(\frac{x}{y})}^{n} = \frac{x^{n}}{y^{n}}$ ; if a quotient is raised to a power, then apply that power to the numerator and the denominator.
power rule for a quotient	${(\frac{x}{y})}^{n} = \frac{x^{n}}{y^{n}}$ ; if a quotient is raised to a power, then apply that power to the numerator and the denominator.
power rule for exponents	${(x^{m})}^{n} = x^{m n}$ ; a power raised to a power can be simplified by multiplying the exponents.
power rule for exponents	${(x^{m})}^{n} = x^{m n}$ ; 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 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	Integers greater than 1 that are divisible only by 1 and itself.
prime number	Integers greater than 1 that are 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 real number, denoted with the symbol $\sqrt{}$ .
principal (nonnegative) square root	The positive square root of a real number, denoted with the symbol $\sqrt{}$ .
product of complex conjugates	The real number that results from multiplying complex conjugates: $(a + b i) (a - b i) = a^{2} + b^{2} .$
product of complex conjugates	The real number that results from multiplying complex conjugates: $(a + b i) (a - b i) = a^{2} + b^{2} .$
product rule for exponents	$x^{m} \cdot x^{n} = x^{m + n}$ ; the product of two expressions with the same base can be simplified by adding the exponents.
product rule for exponents	$x^{m} \cdot x^{n} = x^{m + n}$ ; the product of two expressions with the same base can be simplified by adding the exponents.
Product rule for radicals	$\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ , where a and b represent positive real numbers.
Product rule for radicals	$\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ , where a and b represent positive 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 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 the equality of two ratios.
proportion	A statement of the equality of two ratios.
Pythagorean theorem	Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .
Pythagorean theorem	Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .
Pythagorean theorem	Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .
Pythagorean theorem	Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .
Pythagorean theorem	Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .
Pythagorean theorem	Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .
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 formula	The formula $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , which gives the solutions to any quadratic equation in the form $a x^{2} + b x + c = 0$ , where a, b, and c are real numbers and $a \neq 0$ .
quadratic formula	The formula $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , which gives the solutions to any quadratic equation in the form $a x^{2} + b x + c = 0$ , where a, b, and c are real numbers and $a \neq 0$ .
Quadratic function	A polynomial function with degree 2.
Quadratic function	A polynomial function with degree 2.
quotient	The result after dividing.
quotient	The result after dividing.
quotient rule for exponents	$\frac{x^{m}}{x^{n}} = x^{m - n}$ ; the quotient of two expressions with the same base can be simplified by subtracting the exponents.
quotient rule for exponents	$\frac{x^{m}}{x^{n}} = x^{m - n}$ ; the quotient of two expressions with the same base can be simplified by subtracting the exponents.
Quotient rule for radicals	$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ , where a and b represent positive real numbers.
Quotient rule for radicals	$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ , where a and b represent positive real numbers.
quotients with negative exponents	$\frac{x^{- n}}{y^{- m}} = \frac{y^{m}}{x^{n}}$ , given any integers m and n, where $x \neq 0$ and $y \neq 0$ .
quotients with negative exponents	$\frac{x^{- n}}{y^{- m}} = \frac{y^{m}}{x^{n}}$ , given any integers m and n, where $x \neq 0$ and $y \neq 0$ .
radical	Used when referring to an expression of the form $\sqrt[n]{a}$ .
radical	Used when referring to an expression of the form $\sqrt[n]{a}$ .
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.
radicand	The expression a within a radical sign, $\sqrt[n]{a}$ .
radicand	The expression a within a radical sign, $\sqrt[n]{a}$ .
radicand	The expression a within a radical sign, $\sqrt[n]{a}$ .
radicand	The expression a within a radical sign, $\sqrt[n]{a}$ .
range	The set of second components of a relation. The y-values define the range in relations consisting of points (x, y) in the rectangular coordinate plane.
range	The set of second components of a relation. The y-values define the range in relations consisting of points (x, y) in the rectangular coordinate plane.
rate	A ratio where the units for the numerator and the denominator are different.
rate	A ratio where the units for the numerator and the denominator are different.
ratio	Relationship between two numbers or quantities usually expressed as a quotient.
ratio	Relationship between two numbers or quantities usually expressed as a quotient.
ratio	Relationship between two numbers or quantities usually expressed as a quotient.
ratio	Relationship between two numbers or quantities usually expressed as a quotient.
rational (or fractional) exponents	The fractional exponent m/n that indicates a radical with index n and exponent m: $a^{m / n} = \sqrt[n]{a^{m}}$ .
rational (or fractional) exponents	The fractional exponent m/n that indicates a radical with index n and exponent m: $a^{m / n} = \sqrt[n]{a^{m}}$ .
rational equation	An equation containing at least one rational expression.
rational equation	An equation containing at least one rational expression.
Rational numbers	Numbers of the form $\frac{a}{b}$ , where a and b are integers and b is nonzero.
Rational numbers	Numbers of the form $\frac{a}{b}$ , where a and b are integers and b is nonzero.
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	The reciprocal of a nonzero number n is 1/n.
reciprocal	The reciprocal of a nonzero number n is 1/n.
reciprocals	The reciprocal of a nonzero number n is 1/n.
reciprocals	The reciprocal of a nonzero number n is 1/n.
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.
reducing to lowest terms	Finding equivalent fractions where the numerator and the denominator share no common integer factor other than 1.
reducing to lowest terms	Finding equivalent fractions where the numerator and the denominator share no common integer factor other than 1.
relatively prime	Expressions that share no common factors other than 1.
relatively prime	Expressions that share no common factors other than 1.
remainder	The expression that is left after the division algorithm ends.
remainder	The expression that is left after the division algorithm ends.
restrictions	The set of real numbers for which a rational expression is not defined.
restrictions	The set of real numbers for which a rational expression is not defined.
root	A solution to a quadratic equation in standard form.
root	A solution to a quadratic equation in standard form.
root	A solution to a quadratic equation in standard form.
root	A solution to a quadratic equation in standard form.
round off	A means of approximating decimals with a specified number of significant digits.
round off	A means of approximating decimals with a specified number of significant digits.
run	The horizontal change between any two points on a line.
run	The horizontal change between any two points on a line.
same-side like terms	Like terms of an equation on the same side of the equal sign.
same-side like terms	Like terms of an equation on the same side of the equal sign.
satisfy the equation	After replacing the variable with a solution and simplifying, it produces a true statement.
satisfy the equation	After replacing the variable with a solution and simplifying, it produces a true statement.
scale factor	The reduced ratio of any two corresponding sides of similar triangles.
scale factor	The reduced ratio of any two corresponding sides of similar triangles.
scientific notation	Real numbers expressed in the form $a \times 10^{n}$ , where n is an integer and $1 \leq a < 10$ .
scientific notation	Real numbers expressed in the form $a \times 10^{n}$ , where n is an integer and $1 \leq a < 10$ .
set-builder notation	A system for describing sets using familiar mathematical notation.
set-builder notation	A system for describing sets using familiar mathematical notation.
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.
similar triangles	Triangles with the same shape but not necessarily the same size. The measures of corresponding angles are equal and the corresponding sides are proportional.
similar triangles	Triangles with the same shape but not necessarily the same size. The measures of corresponding angles are equal and the corresponding sides are proportional.
Simple interest	Modeled by the formula $I = p r t$ , where p represents the principal amount invested at an annual interest rate r for t years.
Simple interest	Modeled by the formula $I = p r t$ , where p represents the principal amount invested at an annual interest rate r for t years.
simplified	A radical where the radicand does not consist of any factor that can be written as a perfect power of the index.
simplified	A radical where the radicand does not consist of any factor that can be written as a perfect power 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	Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , then the slope of the line is given by the formula $m = \frac{r i s e}{r u n} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .
slope formula	Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , then the slope of the line is given by the formula $m = \frac{r i s e}{r u n} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .
slope-intercept form	Any nonvertical line can be written in the form $y = m x + b$ , where m is the slope and (0, b) is the y-intercept.
slope-intercept form	Any nonvertical line can be written in the form $y = m x + b$ , 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.
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.
square	The result when the exponent of any real number is 2.
square	The result when the exponent of any real number is 2.
square root	The number that, when multiplied by itself, yields the original number.
square root	The number that, when multiplied by itself, yields the original number.
square root function	The function $f (x) = \sqrt{x}$ .
square root function	The function $f (x) = \sqrt{x}$ .
square root property	For any real number k, if $x^{2} = k$ , then $x = \pm \sqrt{k}$ .
square root property	For any real number k, if $x^{2} = k$ , then $x = \pm \sqrt{k}$ .
squaring property of equality	Given real numbers a and b, where $a = b$ , then $a^{2} = b^{2}$ .
squaring property of equality	Given real numbers a and b, where $a = b$ , then $a^{2} = b^{2}$ .
standard form	A quadratic equation written in the form $a x^{2} + b x + c = 0.$
standard form	A quadratic equation written in the form $a x^{2} + b x + c = 0.$
standard form	Any quadratic equation in the form $a x^{2} + b x + c = 0$ , where a, b, and c are real numbers and $a \neq 0$ .
standard form	Any quadratic equation in the form $a x^{2} + b x + c = 0$ , where a, b, and c are real numbers and $a \neq 0$ .
Strict inequalities	Express ordering relationships using the symbol < for “less than” and > for “greater than.”
Strict inequalities	Express ordering relationships using the symbol < for “less than” and > for “greater than.”
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.
subtracting polynomials	The process of subtracting all the terms of one polynomial from another and combining like terms.
subtracting polynomials	The process of subtracting all the terms of one polynomial from another and combining like terms.
sum of squares	$a^{2} + b^{2}$ does not have a general factored equivalent.
sum of squares	$a^{2} + b^{2}$ does not have a general factored equivalent.
symmetric property	Allows you to solve for the variable on either side of the equal sign, because $5 = x$ is equivalent to $x = 5$ .
symmetric property	Allows you to solve for the variable on either side of the equal sign, because $5 = x$ is equivalent to $x = 5$ .
system of linear inequalities	A set of two or more linear inequalities that define the conditions to be considered simultaneously.
system of linear inequalities	A set of two or more linear inequalities that define the conditions to be considered simultaneously.
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.
Trinomial	Polynomial with three terms.
Trinomial	Polynomial with three terms.
undefined	A quotient such as $\frac{5}{0}$ , which is left without meaning and is not assigned an interpretation.
undefined	A quotient such as $\frac{5}{0}$ , which is left without meaning and is not assigned an interpretation.
Uniform motion	Described by the formula $D = r t$ , where the distance D is given as the product of the average rate r and the time t traveled at that rate.
Uniform motion	Described by the formula $D = r t$ , where the distance D is given as the product of the average rate r and the time t traveled at that rate.
Uniform motion	Described by the formula $D = r t$ , where the distance, D, is given as the product of the average rate, r, and the time, t, traveled at that rate.
Uniform motion	Described by the formula $D = r t$ , where the distance, D, is given as the product of the average rate, r, and the time, t, traveled at that rate.
uniform motion problems	Applications relating distance, average rate, and time.
uniform motion problems	Applications relating distance, average rate, and time.
union	The set formed by joining the individual solution sets indicated by the logical use of the word “or” and denoted with the symbol $\cup$ .
union	The set formed by joining the individual solution sets indicated by the logical use of the word “or” and denoted with the symbol $\cup$ .
unit cost	The price of each unit.
unit cost	The price of each unit.
unlike denominators	Denominators of fractions that are not the same.
unlike denominators	Denominators of fractions that are not the same.
varies inversely	Describes two quantities x and y, where one variable is directly proportional to the reciprocal of the other: $y = \frac{k}{x} .$
varies inversely	Describes two quantities x and y, where one variable is directly proportional to the reciprocal of the other: $y = \frac{k}{x} .$
varies jointly	Describes a quantity y that varies directly as the product of two other quantities x and z: $y = k x z$ .
varies jointly	Describes a quantity y that varies directly as the product of two other quantities x and z: $y = k x z$ .
vertical line	Any line whose equation can be written in the form x = k, where k is a real number.
vertical line	Any line whose equation can be written in the form x = k, where k is a real number.
vertical line test	If a vertical line intersects a graph more than once, then the graph does not represent a function.
vertical line test	If a vertical line intersects a graph more than once, then the graph does not represent a function.
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	$\frac{1}{t_{1}} \cdot t + \frac{1}{t_{2}} \cdot t = 1$ , where $\frac{1}{t_{1}}$ and $\frac{1}{t_{2}}$ are the individual work rates and t is the time it takes to complete the task working together.
work-rate formula	$\frac{1}{t_{1}} \cdot t + \frac{1}{t_{2}} \cdot t = 1$ , where $\frac{1}{t_{1}}$ and $\frac{1}{t_{2}}$ are the individual work rates and t is the time it takes to complete the task working together.
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	$x^{0} = 1$ ; any nonzero base raised to the 0 power is defined to be 1.
zero as an exponent	$x^{0} = 1$ ; any nonzero base raised to the 0 power is defined to be 1.
Zero factor property	Given any real number a, $a \cdot 0 = 0 \cdot a = 0 .$
Zero factor property	Given any real number a, $a \cdot 0 = 0 \cdot a = 0 .$
zero-product property	Any product is equal to zero if and only if at least one of the factors is zero.
zero-product property	Any product is equal to zero if and only if at least one of the factors is zero.

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

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

add polynomials | The process of combining all like terms of two or more polynomials.

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

Additive identity property | Given any real number a, $a + 0 = 0 + a = a .$

Additive inverse property | Given any real number a, $a + (- a) = (- a) + a = 0 .$

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

algebraic fraction | Term used when referring to a rational expression.

Area of a circle | $A = π r^{2}$ , where r represents the radius and the constant $π \approx 3.14$ .

Area of a rectangle | $A = l w$ , where l represents the length and w represents the width.

Area of a square | $A = s^{2}$ , where s represents the length of each side.

Area of a triangle | $A = \frac{1}{2} b h$ , where b represents the length of the base and h represents the height.

Associative property | Given real numbers a, b and c, $(a + b) + c = a + (b + c)$ .

Associative property | Given any real numbers a, b, and c, $(a \cdot b) \cdot c = a \cdot (b \cdot c) .$

Associative property | Given real numbers a, b and c, $(a + b) + c = a + (b + c)$ .

Associative property | Given any real numbers a, b, and c, $(a \cdot b) \cdot c = a \cdot (b \cdot c) .$

asterisk | The symbol (*) that indicates multiplication within text-based applications.

average | Used in reference to the arithmetic mean.

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

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 their equivalent equations, to determine the corresponding value of the other variable.

back substituting | The process of finding the answers to other unknowns after one has been found.

Binomial | Polynomial with two terms.

caret | The symbol ^ that indicates exponents on many calculators, $a^{n} = a^n$ .

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

check by evaluating | We can be fairly certain that we have multiplied the polynomials correctly if we check that a few values evaluate to the same results in the original expression and in the answer.

circumference | The perimeter of a circle given by $C = 2 π r$ , where r represents the radius of the circle and $π \approx 3.14159$ .

collinear | Describes points that lie on the same line.

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

Commutative property | Given real numbers a and b, $a + b = b + a$ .

Commutative property | Given any real numbers a and b, $a \cdot b = b \cdot a .$

Commutative property | Given real numbers a and b, $a + b = b + a$ .

Commutative property | Given any real numbers a and b, $a \cdot b = b \cdot a .$

completing the square | The process of rewriting a quadratic equation in the form ${(x - p)}^{2} = q$ .

complex conjugate | Two complex numbers whose real parts are the same and imaginary parts are opposite. If given $a + b i$ , then its complex conjugate is $a - b i$ .

complex fraction | A fraction where the numerator or denominator consists of one or more fractions.

complex number | Numbers of the form $a + b i$ , where a and b are real numbers.

complex rational expression | A rational expression where the numerator or denominator consists of one or more rational expressions.

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

conjugate binomials | The binomials $(a + b)$ and $(a - b)$ .

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

Constant function | A polynomial function with degree 0.

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

constant term | A term written without a variable factor.

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

cross canceling | Cancelling common factors in the numerator and the denominator of fractions before multiplying.

cross multiplication | If $\frac{a}{b} = \frac{c}{d}$ , then $a d = b c$ .

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

cube root function | The function $f (x) = \sqrt[3]{x}$ .

Cubic function | A polynomial function with degree 3.

decimal | A real number expressed using the decimal system.

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.

dependent system | A system that consists of equivalent equations with 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 as the dependent variable.

difference of cubes | $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}),$ where a and b represent algebraic expressions.

difference of squares | $a^{2} - b^{2} = (a + b) (a - b),$ where a and b represent algebraic expressions.

direct variation | Describes two quantities x and y that are constant multiples of each other: $y = k x$ .

directly proportional | Used when referring to direct variation.

discriminant | The algebraic expression $b^{2} - 4 a c$ .

discriminant | The expression inside the radical of the quadratic formula, $b^{2} - 4 a c$ .

distance formula | Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , calculate the distance d between them using the formula $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$

Distance formula | Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2}),$ calculate the distance d between them using the formula d = $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$

distance formula for a number line | The distance between any two real numbers a and b on a number line can be calculated using the formula $d = | b - a |$ .

distributive property | Given any real numbers a, b, and c, $a (b + c) = a b + a c$ or $(b + c) a = b a + c a$ .

dividend | The numerator of a quotient.

divisor | The denominator of a quotient.

double root | A root that is repeated twice.

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

elimination (or addition) method | A means of solving a system by adding equivalent equations in such a way as to eliminate a variable.

empty set | A subset with no elements, denoted $\emptyset$ or { }.

equality relationship | Express equality with the symbol =. If two quantities are not equal, use the symbol $\neq$ .

equivalent equations | Equations with the same solution set.

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

equivalent inequality | Inequalities 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.

even integers | Integers that are divisible by two or are multiples of two.

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

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

exponential notation | The compact notation $a x^{2} + b x + c = 0.$ used when a factor is repeated multiple times.

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

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

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

factoring a polynomial | The process of rewriting a polynomial as a product of polynomial factors.

Factoring out the GCF | The process of rewriting a polynomial as a product using the GCF of all of its terms.

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

FOIL | When multiplying binomials we apply the distributive property multiple times in such a way as to multiply the first terms, outer terms, inner terms, and last terms.

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

functions | Relations where every x-value corresponds to exactly one y-value. With the definition comes new notation: $f (x) = y$ , which is read “f of x is equal to y.”

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

GCF of a polynomial | The greatest common factor of all the terms of the polynomial.

GCF of monomials | The product of the GCF of the coefficients and all common variable factors.

graph | A point on the number line associated with a coordinate.

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 product of all the common prime factors.

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

guess and check | Used when referring to the trial and error method for factoring trinomials.

horizontal line | Any line whose equation can be written in the form y = k, where k is a real number.

identity | An equation that is true for all possible values.

imaginary numbers | The square roots of any negative real numbers.

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

imaginary unit | Defined as $i = \sqrt{- 1}$ and $i^{2} = - 1$ .

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

inconsistent systems | A system with no simultaneous solution.

independent systems | A system of equations with one ordered pair solution (x, y).

indeterminate | A quotient such as $\frac{0}{0}$ , which is a quantity that is uncertain or ambiguous.

index | The positive integer n in the notation $\sqrt[n]{}$ that is used to indicate an nth root.

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, …}.

interest and money problems | Applications involving simple interest and money.

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 $\cap$ .

interval notation | A textual system of expressing solutions to an algebraic inequality.

inversely proportional | Used when referring to inverse variation.

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

jointly proportional | Used when referring to joint variation.

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

least common denominator (LCD) | The least common multiple of a set of denominators.

least common multiple (LCM) | The smallest number that is evenly divisible by a set of numbers.

line graph | A set of related data values graphed on a coordinate plane and connected by line segments.

linear equation with one variable | An equation that can be written in the general form $a x + b = 0$ , where a and b are real numbers and $a \neq 0$ .

linear equation with two variables | An equation with two variables that can be written in the standard form $a x + b y = c$ , where the real numbers a and b are not both zero.

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

Linear function | A polynomial function with degree 1.

linear inequality | A mathematical statement relating a linear expression as either less than or greater than another.

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 | In this section, we restrict our study to systems of two linear equations with two variables.

literal equations | A formula that summarizes whole classes of problems.

midpoint | Given two points, $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the midpoint is an ordered pair given by $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ .

mixed number | A number that represents the sum of a whole number and a fraction.

Mixture problems | Applications involving a mixture of amounts usually given as a percentage of some total.

Monomial | Polynomial with one term.

Multiplicative identity property | Given any real number a, $a \cdot 1 = 1 \cdot a = a .$

natural (or counting) numbers | The set of counting numbers {1, 2, 3, 4, 5, …}.

negative exponents | $x^{- n} = \frac{1}{x^{n}},$ given any integer n, where x is nonzero.

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

nth root | The number that, when raised to the nth power, yields the original number.

odd integers | Integers that are not divisible by 2.

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 $a - b$ , then the opposite is $- (a - b) = b - a$ .

opposite reciprocals | Two real numbers whose product is −1. Given a real number $\frac{a}{b}$ , the opposite reciprocal is $- \frac{b}{a}$ .

opposite-side like terms | Like terms of an equation on opposite sides of the equal sign.

order | To ensure a single correct result, perform mathematical operations in a specific order.

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

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

parabola | The graph of any quadratic equation $y = a x^{2} + b x + c$ , where a, b, and c are real numbers and $a \neq 0$ .

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

percent | A representation of a number as part of 100: $N % = \frac{N}{100}$ .

perfect cube | The result of cubing an integer.

perfect square | The result of squaring an integer.

perfect square trinomials | The trinomials obtained by squaring the binomials ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ and ${(a - b)}^{2} = a^{2} - 2 a b + b^{2} .$

perimeter | The sum of the lengths of all the outside edges of a polygon.

Perimeter of a rectangle | $P = 2 l + 2 w$ , where l represents the length and w represents the width.

Perimeter of a square | $P = 4 s$ , where s represents the length of a side.

Perimeter of a triangle | $P = a + b + c$ , where a, b, and c each represents the length of a different side.

pie chart | A circular graph divided into sectors whose area is proportional to the relative size of the ratio of the part to the total.

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

point-slope form of a line | Any nonvertical line can be written in the form $y - y_{1} = m (x - x_{1})$ , where m is the slope and $(x_{1}, y_{1})$ 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 long division | The process of dividing two polynomials using the division algorithm.

polynomials with one variable | A polynomial where each term has the form $a_{n} x^{n}$ , where $a_{n}$ 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 $a^{n} = b^{n}$ .

power rule for a product | ${(x y)}^{n} = x^{n} y^{n}$ ; if a product is raised to a power, then apply that power to each factor in the product.

power rule for a quotient | ${(\frac{x}{y})}^{n} = \frac{x^{n}}{y^{n}}$ ; if a quotient is raised to a power, then apply that power to the numerator and the denominator.

power rule for exponents | ${(x^{m})}^{n} = x^{m n}$ ; 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 number | Integers greater than 1 that are 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.

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 positive square root of a real number, denoted with the symbol $\sqrt{}$ .

product of complex conjugates | The real number that results from multiplying complex conjugates: $(a + b i) (a - b i) = a^{2} + b^{2} .$

product rule for exponents | $x^{m} \cdot x^{n} = x^{m + n}$ ; the product of two expressions with the same base can be simplified by adding the exponents.

Product rule for radicals | $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ , where a and b represent positive 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.

proportion | A statement of the equality of two ratios.

Pythagorean theorem | Given any right triangle with legs measuring a and b units and hypotenuse measuring c units, then $a^{2} + b^{2} = c^{2}$ .

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 formula | The formula $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , which gives the solutions to any quadratic equation in the form $a x^{2} + b x + c = 0$ , where a, b, and c are real numbers and $a \neq 0$ .

Quadratic function | A polynomial function with degree 2.

quotient | The result after dividing.

quotient rule for exponents | $\frac{x^{m}}{x^{n}} = x^{m - n}$ ; the quotient of two expressions with the same base can be simplified by subtracting the exponents.

Quotient rule for radicals | $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ , where a and b represent positive real numbers.

quotients with negative exponents | $\frac{x^{- n}}{y^{- m}} = \frac{y^{m}}{x^{n}}$ , given any integers m and n, where $x \neq 0$ and $y \neq 0$ .

radical | Used when referring to an expression of the form $\sqrt[n]{a}$ .

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

radical expression | An algebraic expression that contains radicals.

radicand | The expression a within a radical sign, $\sqrt[n]{a}$ .

range | The set of second components of a relation. The y-values define the range in relations consisting of points (x, y) in the rectangular coordinate plane.

rate | A ratio where the units for the numerator and the denominator are different.

ratio | Relationship between two numbers or quantities usually expressed as a quotient.

rational (or fractional) exponents | The fractional exponent m/n that indicates a radical with index n and exponent m: $a^{m / n} = \sqrt[n]{a^{m}}$ .

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

Rational numbers | Numbers of the form $\frac{a}{b}$ , where a and b are integers and b is nonzero.

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.

reciprocal | The reciprocal of a nonzero number n is 1/n.

reciprocals | The reciprocal of a nonzero number n is 1/n.

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

reducing to lowest terms | Finding equivalent fractions where the numerator and the denominator share no common integer factor other than 1.

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

remainder | The expression that is left after the division algorithm ends.

restrictions | The set of real numbers for which a rational expression is not defined.

root | A solution to a quadratic equation in standard form.

round off | A means of approximating decimals with a specified number of significant digits.

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

same-side like terms | Like terms of an equation on the same side of the equal sign.

satisfy the equation | After replacing the variable with a solution and simplifying, it produces a true statement.

scale factor | The reduced ratio of any two corresponding sides of similar triangles.

scientific notation | Real numbers expressed in the form $a \times 10^{n}$ , where n is an integer and $1 \leq a < 10$ .

set-builder notation | A system for describing sets using familiar mathematical notation.

similar radicals | Term used when referring to like radicals.

similar terms | Used when referring to like terms.

similar triangles | Triangles with the same shape but not necessarily the same size. The measures of corresponding angles are equal and the corresponding sides are proportional.

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

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

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.

slope formula | Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , then the slope of the line is given by the formula $m = \frac{r i s e}{r u n} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

slope-intercept form | Any nonvertical line can be written in the form $y = m x + b$ , 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 to a linear inequality | A real number that produces a true statement when its value is substituted for the variable.

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.

square | The result when the exponent of any real number is 2.

square root | The number that, when multiplied by itself, yields the original number.

square root function | The function $f (x) = \sqrt{x}$ .

square root property | For any real number k, if $x^{2} = k$ , then $x = \pm \sqrt{k}$ .

squaring property of equality | Given real numbers a and b, where $a = b$ , then $a^{2} = b^{2}$ .

standard form | A quadratic equation written in the form $a x^{2} + b x + c = 0.$

standard form | Any quadratic equation in the form $a x^{2} + b x + c = 0$ , where a, b, and c are real numbers and $a \neq 0$ .

Strict inequalities | Express ordering relationships using the symbol < for “less than” and > for “greater than.”

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.

subtracting polynomials | The process of subtracting all the terms of one polynomial from another and combining like terms.

sum of squares | $a^{2} + b^{2}$ does not have a general factored equivalent.

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

system of linear inequalities | A set of two or more linear inequalities that define the conditions to be considered simultaneously.

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.

Trinomial | Polynomial with three terms.

undefined | A quotient such as $\frac{5}{0}$ , which is left without meaning and is not assigned an interpretation.

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

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

uniform motion problems | Applications relating distance, average rate, and time.

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

unit cost | The price of each unit.

unlike denominators | Denominators of fractions that are not the same.

varies inversely | Describes two quantities x and y, where one variable is directly proportional to the reciprocal of the other: $y = \frac{k}{x} .$

varies jointly | Describes a quantity y that varies directly as the product of two other quantities x and z: $y = k x z$ .

vertical line | Any line whose equation can be written in the form x = k, where k is a real number.

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

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 formula | $\frac{1}{t_{1}} \cdot t + \frac{1}{t_{2}} \cdot t = 1$ , where $\frac{1}{t_{1}}$ and $\frac{1}{t_{2}}$ are the individual work rates and t is the time it takes to complete the task working together.

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

zero as an exponent | $x^{0} = 1$ ; any nonzero base raised to the 0 power is defined to be 1.

Zero factor property | Given any real number a, $a \cdot 0 = 0 \cdot a = 0 .$

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

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