Algebra and Trigonometry 2e

# 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, $a+( b+c )=( a+b )+c a+( b+c )=( a+b )+c$
associative property of multiplication
the product of three numbers may be grouped differently without affecting the result; in symbols, $a⋅( b⋅c )=( a⋅b )⋅c a⋅( b⋅c )=( a⋅b )⋅c$
base
in exponential notation, the expression that is being multiplied
binomial
a polynomial containing two terms
coefficient
any real number $a i a i$ in a polynomial in the form $a n x n +...+ a 2 x 2 + a 1 x+ a 0 a n x n +...+ a 2 x 2 + a 1 x+ a 0$
commutative property of addition
two numbers may be added in either order without affecting the result; in symbols, $a+b=b+a a+b=b+a$
commutative property of multiplication
two numbers may be multiplied in any order without affecting the result; in symbols, $a⋅b=b⋅a a⋅b=b⋅a$
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, $a⋅( b+c )=a⋅b+a⋅c a⋅( b+c )=a⋅b+a⋅c$
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 $a x 2 +bx+c a x 2 +bx+c$ 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, $a+0=a a+0=a$
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, $a⋅1=a a⋅1=a$
index
the number above the radical sign indicating the nth root
integers
the set consisting of the natural numbers, their opposites, and 0: ${ …,−3,−2,−1,0,1,2,3,… } { …,−3,−2,−1,0,1,2,3,… }$
inverse property of addition
for every real number $a, a,$ there is a unique number, called the additive inverse (or opposite), denoted $−a, −a,$ which, when added to the original number, results in the additive identity, 0; in symbols, $a+( −a )=0 a+( −a )=0$
inverse property of multiplication
for every non-zero real number $a, a,$ there is a unique number, called the multiplicative inverse (or reciprocal), denoted $1 a , 1 a ,$ which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, $a⋅ 1 a =1 a⋅ 1 a =1$
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
the coefficient of the 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: ${ 1,2,3,… } { 1,2,3,… }$
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 $a a$ that when raised to the nth power equals $a a$
principal square root
the nonnegative square root of a number $a a$ that, when multiplied by itself, equals $a a$
the symbol used to indicate a root
an expression containing a radical symbol
the number under the radical symbol
rational expression
the quotient of two polynomial expressions
rational numbers
the set of all numbers of the form $m n , m n ,$ where $m m$ and $n n$ are integers and $n≠0. n≠0.$ 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 $a× 10 n a× 10 n$ where $1≤| a |<10 1≤| a |<10$ and $n n$ is an integer
term of a polynomial
any $a i x i a i x i$ of a polynomial in the form $a n x n +...+ a 2 x 2 + a 1 x+ a 0 a n x n +...+ a 2 x 2 + a 1 x+ a 0$
trinomial
a polynomial containing three terms
variable
a quantity that may change value
whole numbers
the set consisting of 0 plus the natural numbers: ${ 0,1,2,3,… } { 0,1,2,3,… }$
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