1. Prerequisite Flashcards
A combination of numbers, variables, and operation symbols is called an algebraic ______.
expression
If n is a counting number, b^n, read ________, indicates that there are n factors of b. The number b is called the _____ and the number n is called the _____.
b to the nth power
base
exponent
An equation that expresses a relationship between two or more variables, such as H = 9/10(220 - a), is called a/an ______. The process of finding such equations to describe real-world phenomena is called mathematical ______. Such equations, together with the meaning assigned to the variables, are called mathematical _______.
formula
modeling
model
The set of elements common to both set A and set B is called the ______ of sets A and B, and is symbolized by ______.
intersection
a ∩ b
The set of elements that are members of set A or set B or of both sets is called the _____ of sets A and B and is symbolized by ______.
union
a ∪ b
The set {1, 2, 3, 4, 5, …} is called the set of _____ numbers.
natural
The set {0, 1, 2, 3, 4, 5, …} is called the set of _____ numbers.
whole
The set {…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …} is called the set of _____.
integers
The set of numbers in the form a/b, where a and b belong to the set {…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …} and b ≠ 0, is called the set of _____ numbers.
rational
The set of numbers whose decimal representations are neither terminating nor repeating is called the set of _____ numbers.
irrational
Every real number is either a/an _____ number or a/an _____ number.
rational
irrational
The notation |x| is read the _____ of x. If x ≥ 0, then |x| = ______. If x < 0, then |x| = _____.
absolute value
x
–x
The commutative properties state that a + b = _____ and ab = ______.
b + a
ba
The associative properties state that (a + b) + c = _____ and _____ = a(bc).
a + (b + c)
(ab)c
The distributive property states that a(b + c) = _____.
ab + ac
a + (–a) = _____: The sum of a real number and its additive _____ is _____, the additive _____.
0
inverse
0
identity
a ∙ 1/a = 1, a ≠ 0: The product of a nonzero real number and its multiplicative _____ is _____, the multiplicative _____.
inverse
1
identity
An algebraic expression is _____ when the parentheses have been removed and like terms have been combined.
simplified
–(–a) = _____.
a
The product rule for exponents states that b^m ∙ b^n = _____. When multiplying exponential expressions with the same base, _____ the exponents.
b^(m + n)
add
The quotient rule for exponents states that (b^m)/(b^n) = _____, b ≠ 0. When dividing exponential expressions with the same nonzero base, _____ the exponents.
b^(m – n)
subtract
If b ≠ 0, then b^0 = _____.
1
The negative-exponent rule states that b^–n = ______, b ≠ 0.
1 / b^n
True or false: 5^–2 = –5^2
false
Negative exponents in denominators can be evaluated using 1/(b^–n) = _____, b ≠ 0.
b^n
True or false: 1/(8^–2) = 8²
true
A positive number is written in scientific notation when it is expressed in the form a × 10^n, where a is _____ and n is a/an _____.
a number greater than or equal to 1 and less than 10
integer
True or false: 4 × 10^3 is written in scientific notation.
true
True or false: 40 × 10² is written in scientific notation.
false
The symbol √ is used to denote the nonnegative or _____, square root of a number.
principal
√64 = 8 because _____ = 64.
8²
√a² = _____
|a|
The product rule for square roots states that if a and b are nonnegative, then √ab = _____.
√a ∙ √b
The quotient rule for square roots states that if a and b are nonnegative and b ≠ 0, then √(a/b) = ______.
√a / √b
8√3 + 10√3 = _____
18√3
√3 + √75 = √3 + √(25 ∙ 3) = √3 + _____√3 = ______
5
6√3
The conjugate of 7 + √3 is _____
7 – √3
We rationalize the denominator of 5 / (√10 – √2) by multiplying the numerator and denominator by _____.
√10 + √2
In the expression ∛64, the number 3 is called the _____ and the number 64 is called the _____.
index
radicand
^5√–32 = –2 because _____ = –32
(–2)^5
If n is odd, ^n√(a^n) = _____
a
if n is even, ^n√(a^n) = _____
|a|
a^(1/n) = _____
^n√a
A polynomial is a single term or the sum of two or
more terms containing variables with exponents that are ______ numbers.
whole
It is customary to write the terms of a polynomial in
the order of descending powers of the variable. This is called the ______ form of a polynomial.
standard
A simplified polynomial that has exactly one term is
called a/an ______.
monomial
A simplified polynomial that has two terms is called
a/an ______.
binomial
A simplified polynomial that has three terms is called
a/an ______.
trinomial
If a ≠ 0, the degree of ax^n is ______.
n
Polynomials are added by combining ______ terms.
like
To multiply 7x^3(4x^5 – 8x² + 6), use the ______ property to multiply each term of the trinomial ______ by the monomial ______.
distributive
4x^5 – 8x² + 6
7x^3
To multiply (5x + 3)(x² + 8x + 7), begin by multiplying each term of x² + 8x + 7 by ______. Then
multiply each term of x² + 8x + 7 by ______. Then
combine ______ terms.
5x
3
like
When using the FOIL method to find (x + 7)(3x + 5), the product of the first terms is ______, the product of the outside terms is ______, the product of the inside
terms is ______, and the product of the last terms
is ______.
3x²
5x
21x
35
(A + B)(A – B) = ______. The product of the
sum and difference of the same two terms is the square of the first term ______ the square of the second term.
A² – B²
minus
(A + B)² = ______. The square of a binomial sum is the first term ______ plus 2 times the _______ plus the last term ______.
A² + 2AB + B²
squared
product of the terms
squared
(A – B)² = ______. The square of a binomial
difference is the first term squared ______ 2 times
the ______ _______ the last term squared.
A² – 2AB + B²
minus
product of the terms
plus
If a ≠ 0, the degree of (ax^n)(y^m) is _______.
n + m
Which technique should we use for factoring the polynomial?
16x² – 25
Factoring the difference of two squares :
A² – B² = (A + B)(A – B)
Which technique should we use for factoring the polynomial?
27x^3 – 1
Factoring the difference of two cubes
A^3 – B^3 = (A – B)(A² + AB + B²)
Which technique should we use for factoring the polynomial?
x² + 7x + xy + 7y
Factoring by grouping
Which technique should we use for factoring the polynomial?
4x² + 8x + 3
Factoring trinomials by trial and error
Which technique should we use for factoring the polynomial?
9x² + 24x + 16
Factoring perfect square trinomials
A² + 2AB + B² = (A + B)²
A² – 2AB + B² = (A – B)²
Which technique should we use for factoring the polynomial?
5x² + 10x
Factoring out the GCF
Which technique should we use for factoring the polynomial?
x^3 + 1000
Factoring the sum of two cubes
A^3 + B^3 = (A + B)(A² – AB + B²)
The algebraic expression (x + 1)^(1/2) – 1/3(x + 1)^(3/2) can be factored using ______ as the greatest denominator.
(x + 1)^(1/2)
A rational expression is the quotient of two ______.
polynomials
The set of real numbers for which a rational expression
is defined is the ______ of the expression. We must
exclude all numbers from this set that make the
denominator of the rational expression ______.
domain
0
We simplify a rational expression by ______ the
numerator and the denominator completely. Then we divide the numerator and the denominator by
any ______.
factoring
common factors
(x/5) ∙ (x/3) = _______
x²/15
(x/5) ÷ (x/3) = _______, x ≠ 0
3/5
x²/3 – (x – 2)/3 = ______
(x² – x + 2) / 3
Consider the following subtraction problem:
(x – 1) / (x² + x – 6) – (x – 2) / (x² + 4x + 3)
The factors of the first denominator are ______.
The factors of the second denominator are ______.
The LCD is _______.
x + 3 and x – 2
x + 3 and x + 1
(x + 3)(x – 2)(x + 1)
An equivalent expression for (3x + 2) / (x – 5) can be obtained by multiplying the numerator and denominator by _____.
3x + 4
A rational expression whose numerator or denominator orboth contain rational expressions is called a/an ______ rational expression or a/an ______ fraction.
complex
complex
