Key Terms

Key Terms

algebraic expression

constants and variables combined using addition, subtraction, multiplication, and division

associative property of addition

the sum of three numbers may be grouped differently without affecting the result; in symbols, $a+\left(b+c\right)=\left(a+b\right)+c$

associative property of multiplication

the product of three numbers may be grouped differently without affecting the result; in symbols, $a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c$

base

in exponential notation, the expression that is being multiplied

binomial

a polynomial containing two terms

coefficient

any real number $a_i$ in a polynomial in the form $a_n{x}^n+\mathrm{...}+a_2{x}^2+a_{1}x+a_0$

commutative property of addition

two numbers may be added in either order without affecting the result; in symbols, $a+b=b+a$

commutative property of multiplication

two numbers may be multiplied in any order without affecting the result; in symbols, $a\cdot b=b\cdot a$

constant

a quantity that does not change value

degree

the highest power of the variable that occurs in a polynomial

difference of squares

the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign

distributive property

the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$

equation

a mathematical statement indicating that two expressions are equal

exponent

in exponential notation, the raised number or variable that indicates how many times the base is being multiplied

exponential notation

a shorthand method of writing products of the same factor

factor by grouping

a method for factoring a trinomial in the form $a{x}^2+bx+c$ by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression

formula

an equation expressing a relationship between constant and variable quantities

greatest common factor

the largest polynomial that divides evenly into each polynomial

identity property of addition

there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, $a+0=a$

identity property of multiplication

there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, $a\cdot 1=a$

index

the number above the radical sign indicating the nth root

integers

the set consisting of the natural numbers, their opposites, and 0: $\left\{\dots ,\mathrm{-3},\mathrm{-2},\mathrm{-1},0,1,2,3,\dots \right\}$

inverse property of addition

for every real number $a,$ there is a unique number, called the additive inverse (or opposite), denoted $-a,$ which, when added to the original number, results in the additive identity, 0; in symbols, $a+\left(-a\right)=0$

inverse property of multiplication

for every non-zero real number $a,$ there is a unique number, called the multiplicative inverse (or reciprocal), denoted $\frac{1}{a},$ which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, $a\cdot \frac{1}{a}=1$

irrational numbers

the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers

leading coefficient

the coefficient of the leading term

leading term

the term containing the highest degree

least common denominator

the smallest multiple that two denominators have in common

monomial

a polynomial containing one term

natural numbers

the set of counting numbers: $\left\{1,2,3,\dots \right\}$

order of operations

a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations

perfect square trinomial

the trinomial that results when a binomial is squared

polynomial

a sum of terms each consisting of a variable raised to a nonnegative integer power

principal nth root

the number with the same sign as $a$ that when raised to the nth power equals $a$

principal square root

the nonnegative square root of a number $a$ that, when multiplied by itself, equals $a$

radical

the symbol used to indicate a root

radical expression

an expression containing a radical symbol

radicand

the number under the radical symbol

rational expression

the quotient of two polynomial expressions

rational numbers

the set of all numbers of the form $\frac{m}{n},$ where $m$ and $n$ are integers and $n\ne 0.$ Any rational number may be written as a fraction or a terminating or repeating decimal.

real number line

a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.

real numbers

the sets of rational numbers and irrational numbers taken together

scientific notation

a shorthand notation for writing very large or very small numbers in the form $a\times {10}^n$ where $1\le \left|a\right|<10$ and $n$ is an integer

term of a polynomial

any $a_i{x}^i$ of a polynomial in the form $a_n{x}^n+\mathrm{...}+a_2{x}^2+a_{1}x+a_0$

trinomial

a polynomial containing three terms

variable

a quantity that may change value

whole numbers

the set consisting of 0 plus the natural numbers: $\left\{0,1,2,3,\dots \right\}$