Algebra and Trigonometry

# Key Terms

algebraic expression
constants and variables combined using addition, subtraction, multiplication, and division
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$a n x n +...+ a 2 x 2 + a 1 x+ a 0 a n x n +...+ a 2 x 2 + a 1 x+ a 0$
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
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
integers
the set consisting of the natural numbers, their opposites, and 0:${ …,−3,−2,−1,0,1,2,3,… } { …,−3,−2,−1,0,1,2,3,… }$
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:${ 0,1,2,3,… } { 0,1,2,3,… }$