Variable
This is a symbol (generally a letter) that is used to represent an unknown number or any one of several numbers.
Rational Numbers
This is any number that can be written in the form a/b where a and b are integers and b?0.
(The letter ? represents the whole set)
Rational Number Examples
23,71,?53,2710, and 3?10
Each is in the form ab where a and b are integers and b?0.
13/4 and 2.33 are also rational numbers. They are not in the form ab, but they can be written in that form:
13/4=7/4 and 2.33=233100=233100
Irrational Numbers
This is any number that can be written as an infinite, non-repeating decimal. These cannot be written in fraction form.
Irrational Number Examples
a. ?=3.14159265358979... ?
has no repeating pattern in its decimal form.
b. 2?=1.414213562...
The square root of 2 has no repeating pattern in its decimal form.
c. e=2.718281828459045... e is a number used in higher mathematics and engineering courses.
d.
Number Examples
?3 : integer, rational number, real number
12 : rational number, real number
4 : natural number, whole number, integer, rational number, real number
10??? : irrational number, real number
Number Examples 2
integers.
Solution: ?2 and 0 are integers.
rational numbers.
Solution: ?2,?1.1,?12,0,58, and 1.7
irrational numbers.
Solution: ?3? and 1.7???? are irrational numbers.
?3? is approximately ?1.732 and 1.7???? is approximately 1.304.
0
Rational Number, Integer, Whole Number
Absolute Value Examples
Example 1: Absolute Value
|5|=5
|?6|=?(?6)=6
|?|=?
|0|=0
????34???=?(?34)=34
?|?5.2|=?5.2
If |x|=6, what are the possible values for x?
x=6 or x=?6 because both |6|=6 and |?6|=6.
We say that {?6,6} is the solution set of the equation.
If |x|=?4, what are
Natural Numbers
These are counting numbers starting with the number 1. {1,2,3,4,5,6,...}
(The letter ? represents the whole set)
Whole Numbers
These are counting numbers starting with the number 0. {0,1,2,3,4,5,6,...}
(The letter ?? represents the whole set)
Integers
These are numbers that can be represented on a number line, no decimals. {..., ?4,?3,?2,?1,0,1,2,3,4,...}
(The letter ? represents the whole set)
Real numbers
Formed with the set of rational and irrational numbers.
(The letter ? represents the whole set)
Natural number relationships
Every one of these numbers is also a:
-whole number
-integer
-rational number
-real number
Whole number relationships
Every one of these numbers is also a:
-integer
-rational number
-real number
Integer relationships
Every one of these numbers is also a:
-rational number
-real number
Rational number relationships
Every one of these numbers is also a:
-real number
Irrational number relationships
Every one of these numbers is also a:
-real number
Real number line
Every rational and irrational number has a corresponding point on this line.
Symbols for Multiplication
Symbol Description Example
?
raised dot 4?7
( )
numbers inside or next to parentheses
5(10) or (5)10 or (5)(10)
�
cross sign 6�12 or 12�6������
number written next to variable
variable written next to variable
8x
xy
Rules for Dividing Positive and Negative Real Numbers
For positive real numbers a and b,
The quotient of two positives is positive: ab=+ab.
The quotient of two negatives is positive: ?a?b=+ab.
The quotient of a positive and a negative is negative: ?ab=a?b=?ab.
In summary:
The quotient of numbers with like si
Order of Operations
PEMDAS (Please excuse my Dear Aunt Sally)
1. Parethesis (groups)
2. Exponentials
3. Multiplication and Division, left to right
4. Addition and Subtraction, left to right
rules for order of operations
Simplify within symbols of inclusion (parentheses, brackets, braces, fraction bar, absolute value bars) beginning with the innermost symbols.
Find any powers indicated by exponents or roots.
Multiply or divide from left to right.
Add or subtract from left
16?3�23?(18+20)
Solution:
16?3�=====23?(18+20)16?3�23?3816?3�8?3848�8?386?38?32Add within the parentheses.Evaluate the exponents.Multiply or divide from left to right.Add or subtract from left to right.
?
an element of
?
such that
{x?x is an even integer}
the set of all x such that x is an even integer
<
less than
>
greater than
?
less than or equal to
?
greater than or equal to
{x?x??}
the set of all x such that x is an element of the set of integers
term
an expression that involves only multiplication and/or division with constants and/or variables.
Examples: 2x?, �/?x�y, 14, 5.6a
Conditional Equation
An equation with a finite number of solutions
Identity (Equation)
An equation with an infinite number of solutions
Contradiction (Equation)
An equation with no solutions
Applications (aka)
Word Problems (aka)
Formula for Distance
D=RT (Distance = Rate � Time)
Formula for Profit
P=S-C (Profit = Selling Price - Cost)
Formula for Simple Interest
I=PRT (Interest = Principle � Rate � Time)
I=PR (Interest = Principle � Rate) if just one year
Solve Equalities
What is done to one side of the equation must be done to the other
Solve Inequalities
--Multiply or Divide by a negative number results in an OPPOSITE inequality sign.
--All else keeps the inequality same the same.
Slope-Intercept Form
(equation)
y=mx+b
(m is the slope and the y-intercept is (0,b))
Slope of the line
(equation)
(Slope (m) = Change in Y � Change in X = Rise � Run and x? ? y?)
Horizontal line
(0,#)
y=b
slope is 0
Vertical line
(#,0)
x=a
slope is 0
Slope
Rise � Run
Point-Slope Form
(equation)
y-y?=m(x-x?)
Standard Form
(equation)
Ax+By=C
Parallel lines
Slope is identical
y=2x+1
y=2x+3
Perpendicular lines
Slope is the negative reciprocal
y=�x+1
y=-2x-3
Relation
Set of ordered pairs of real numbers
Domain (D)
The ___ of a relation is the set of all first coordinates in the relation.
Range (R)
The ___ of a relation is the set of all second coordinates in the relation.
Domain axis
The horizontal axis (x-axis) is called this
Range axis
The vertical axis (y-axis) is called this
Function (�)
-A relation in which each domain element has exactly one corresponding range element.
-A relation in which each first coordinate appears only once.
-A relation in which no two ordered pairs have the same first coordinate.
Vertical line test
if any vertical line intersects the graph of a relation at more than one point, then the relation is NOT a function
Linear function
A function represented by an equation of the form y=mx+b. The domain is the set of all real numbers DE=(-?,?)
Nonlinear function
y=2�x-1 (x?1)
D=(-?,1)?(1,?)
?
(A?B)
union symbol: meaning that the set belongs to either or to both. Think "or
?
(A?B)
intersection symbol: meaning all elements belong to both. Think "and
function notation
(formula)
�(x)=mx+b
System of equations
Set of simultaneous equations
Two or more linear equations considered at one time.
Consistent
two linear equations in two variables:
have exactly one solution (the lines intersect at one point)
Inconsistent
two linear equations in two variables:
have no solution (the lines are parallel)
Dependent
two linear equations in two variables:
have an infinite number of solutions (the same line)
Product Rules for Exponents
Add the exponents together
Quotient Rule for Exponents
(top exponent) minus (bottom exponent)
Rule for Negative Exponents
Reciprocal of the base.
Power Rule
Multiply the exponents
Power Rule for Products
Raise each factor to that power
Power Rule for Quotients (Fractions)
Raise the numerator and denominator to that power
Exponent Zero (0)
a?=1
0?=undefined
Exponent One (1)
a�=a
Negative base number without parenthesis
-x� = (-1)x�
Without parenthesis the base is not negative but is preceded by a -1
Negative base number in parenthesis
(-x)� = (-x)(-x)
With parenthesis the base is negative
Monomial
-n is a whole number
-k is a real number
-No variables or positive exponents in the denominator.
-No negative exponents in the numerator.
-No fractional exponents (only whole #s)
Degree of the monomial
n is called this
With multiple exponents - add them together.
Coefficient of the monomial
k is called this
Polynomial
monomial or the algebraic sum or difference of monomials
Degree of a polynomial
The largest of the degrees of its term after like terms have been combined.
With multiple exponents - use the largest one AFTER all like term are combined.
Leading coefficient (Coefficient of the polynomial)
The coefficient of the term of the largest degree.
(aka the coefficient of the term with the largest exponent)
Polynomial with *one* term
(after simplification)
Monomial
Example: 15x�
(3rd-degree monomial)
Polynomial with *two* terms
(after simplification)
Binomial
Example: 4x-10
(1st-degree binomial)
Polynomial with *three* terms
(after simplification)
Trinomial
Example: -x?+2x-1
(4th-degree trinomial)
Linear polynomial
Degree 0 or 1
Example: 3x+8
(1st-degree binomial, leading coefficient 3)
Quadratic polynomial
Degree 2
Example: 5x�-3x+7
(2nd-degree trinomial, leading coefficient 5)
Cubic polynomial
Degree 3
Example: -10y�
(3rd-degree monomial, leading coefficient -10)
Adding polynomials
Combine like terms
(written with highest degree aka. exponent first)
Subtracting polynomials
-Change the signs of the term being subtracted (positive to negative and vice versa)
-Combine like terms
(written with highest degree aka. exponent first)
Evaluation of Polynomials
P is the polynomial and x is the variable used in the polynomial.
P(3) means 3 is substituted for all x's
Distributive Property
Difference of Two squares
Sum of the same binomial terms *but* one is addition and one subtraction.
Perfect Square Trinomials
(Square of a binomial sum)
1. the first term is squared (first of the trinomial)
2. the coefficients of the two are multiplied and then doubled (second of the trinomial)
3. the second term is squared (third of the trinomial)
Perfect Square Trinomials
(Square of a binomial difference)
the middle term is doubled
Squares of binomial - Common Error
FOIL Method
*F* irst terms
*O* utside terms
*I* nside terms
*L* ast terms
Factoring out the GCF
(find a monomial that is the GCF of a polynomial)
1. Find the variable(s) of highest degree and the largest integer coefficient that is a factor of each term of the polynomial. (This is one factor)
2. Divide this monomial factor into each term of the polynomial resulting in another polynomial factor.
(if
Not factorable
(irreducible or prime)
Polynomials that do not have a GCF (greatest common factor)
Second-degree trinomials
(in the variable x)
ax�+bx+c
(a, b, and c are real constants)
Factor trinomial with leading coefficient 1
Find two factors of the constant term whose sum in the coefficient of the middle term.
(if they do not exist, the trinomial is not factorable)
Example: x�+11x+30 = (x+5)(x+6)
Factor trinomial with leading coefficient other than 1
Use the FOIL method, followed by trial and error.
-Find all various combinations of *F* and *L*
-List all possible combinations in their respective *F* and *L* positions.
-Only check the sums in the *O* and *I* positions, until you find the sum of the mid
ac-Method of Factoring trinomials
ax�+bx+c
1.Multiply *a* & *c*
2.Find two integers whose product is *ac* and whose sum is *b*. (if none, then not factorable)
3.Rewrite the middle term *bx* using the two numbers from above as coefficients.
4.Factor by grouping the 1st two terms and the la
11�
121
12�
144
13�
169
14�
196
15�
225
16�
256
17�
289
18�
324
19�
361
20�
400
(x+a)(x-a)
x�-a�
(x+a)�
x�+2ax+a�
(x-a)�
x�-2ax+a�
Not factorable squares
x�+a � = *Not* factorable
(x-a)� = Factorable
Factoring Perfect Square Trinomials
In a perfect square trinomial, both the first and last terms of the trinomial must be perfect squares. If the first term is of the form x� and the last term is of the form a�, then the middle term must be of the form 2ax or ?2ax.
x�+2ax+a� = (x+a)�
x�-2ax
Finding the Roots
Finding the Solutions to an equation
Quadratic equations
ax�+bx+c=0
a, b, and c are constants and a?0
(if a=1, then the answer is the 2 factors whose product is *c* and sum is *b*, check your positive/negative signs)
Zero-factor Property
if ab = 0, then a=0 or b=0 or both.
Before factoring out anything, get everything one one side of the equals side so it equals 0
Double root
(root of multiplicity two)
If you have a double factor, then the solution is called ____.
Equation: (x+1)(x+1)=0
Answer = -1 (doubled)
Factor Theorem
If x=c is a root of a polynomial equation in the form P(x)=0, then x-x is a factor of the polynomial P(x).
Consecutive Integers
Integers are _______ if each is 1 more than the previous integer. Three of these can be represented by n, n+1, n+2.
Example: 5,6,7
Consecutive Even Integers
These integers are _______ if each is 2 more than the previous even integer. Three of these can be represented by n, n+2, n+4.
Example: 24, 26, 28
Consecutive Odd Integers
These integers are _______ if each is 2 more than the previous odd integer. Three of these can be represented by n, n+2, n+4.
Examples: 41,43,45
Pythagorean Theorem
c�=a�+b�
Hypotenuse
The longest side of a right triangle.