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)
Irrational Numbers
This is any number that can be written as an infinite, non-repeating decimal. These cannot be written in fraction form.
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.
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
?
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 sign 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
Closed interval notation
[ ] means includes the numbers listed
Example: [-3,2] means -3?x?2
Solid dot on a number line
Open interval notation
( ) means it excludes the number listed
Example (-1,4) means -1<x<4
open dot on the number line
the infinity symbol is always open in parenthesis
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 of a Product
Raise each factor to that power
Power Rule of a Quotient (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 more than one variable, this is sum of the exponents of its variable
A nonzero constant (such as 5) is considered to be monomial of degree 0.
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, stays positive (second of the trinomial)
3. the second term is squared (third of the trinomial)
Perfect Square Trinomials
(Square of a binomial difference)
1. the first term is squared (first of the trinomial)
2. the coefficients of the two are multiplied and then doubled, stays negative (second of the trinomial)
3. the second term is squared (third of the trinomial)
Squares of binomial - Common Error
FOIL Method
*F* irst terms
*O* utside terms
*I* nside terms
*L* ast terms
Dividing by a Monomial
(not covered in the course)
Divide each term in the numerator by the monomial in the denominator and simply each fraction.
(not covered in the course)
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)
Factoring by grouping
Group polynomials with four or more terms in such a way that a common binomial factor or some other form of factors can be recognized.
Example: 2x�-20x-3x+30 = 2x(x-10)-3(x-10) = (x-10)(2x-3)
Always write the GCF first
Factor trinomial with leading coefficient 1
x�+bx+c
Find two factors of *c* whose sum is *b*.
(if they do not exist, the trinomial is not factorable)
Example: x�+11x+30 = (x+5)(x+6)
Trial-and-Error Method of Factoring with FOIL
(leading coefficient other than 1)
ax�+bx+c
-Find all possible combinations of factors of *ax�* and *c* in their respective *F* and *L* positions.
-Only check the sums in the *O* and *I* positions, until you find the sum to be *bx* (the middle term).
(if they do not exist, the trinomial is
ac-Method of Factoring trinomials
(leading coefficient other than 1)
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
Difference of two squares
x�-a� = (x+a)(x-a)
The first binomial factor is addition, the second is subtraction.
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�
The square of the hypotenuse is equal to the sum of the squares of the legs.
Hypotenuse
The longest side of a right triangle.
Rational Expression
If the denominator becomes 0 because of substituting a variable, then the expression is undefined.
Restrictions on the variable.
The undefined value of the variable.
Fundamental Principle of Fractions
Reciprocal of a fraction
Flip the fraction upside down. Multiplying a fraction by its reciprocal = 1
Multiplication of a fraction
Multiply all the numerators, multiply all the denominators.
Division of a fraction
Copy, Dot, Flip"
Multiply the first fraction by the reciprocal of the second.
Addition of a fraction
The denominators must match, then add the denominators together
Subtraction of a fraction
The denominators must match, then subtract the second denominator from the first.
Opposites in Rational Expressions
When nonzero opposites are divided, the quotient is always -1.
Common Error in Rational Expressions
Multiply Rational Expressions
1. Completely factor each numerator and denominator.
2. Multiply the numerators, multiply the denominators, keeping it in a factored form.
3. "Divide Out" any common factors, no denominator can be 0.
Divide Rational Expressions
1. Copy. Dot. Flip
2. Completely factor each numerator and denominator.
3. Multiply the numerators, multiply the denominators, keeping it in a factored form.
4. "Divide Out" any common factors, no denominator can be 0.
Add Rational Expressions
1. Completely factor each numerator and denominator.
2. Add the numerators
3. Keep the common denominators
LCM for a set of polynomials
1. Completely factor each polynomial (including prime factors for numerical factors)
2. Form the product of all factors that appear, using each factor the most number of times it appears in any one polynomial
Adding Rational expressions with different denominators
1. Find the LDC (the LCM of the denominators)
2. Rewrite each fraction in an equivalent form with the LCD as the denominator
3. Add the numerators and keep the common denominator
4. Reduce if possible
Placement of Negative Signs
Subtraction with Rational Expressions
Common Error when subtracting rational expressions
Ratios
Proportion
An equation stating that two ratios are equal
Word problems with proportions
Use the correct ration of units on both sides. One of the following conditions must be true:
1. Numerators agree in type AND denominators agree in type
-or-
2. Numerators correspond AND denominators correspond
Similar triangles
Must meet BOTH conditions
1. The corresponding angles are equal
2. The corresponding sides are proportional
Solve an Equation containing rational expressions
1. Find the LCM of the denominators
2. Multiply both sides of the equation by the LCM and simplify
3. Solve the resulting equation (this will have only polynomials on both sides)
4. Check each solution in the original equation (any solution with a 0 denom
Word problems related to hours worked
A man can dig a ditch in 3 hours, what can he do in 1 hour = 1/3
Equation for Distance, Rate, and Time
d=rt
Finding the square root
Radical sign
Radicand
The number under the Radical sign
Radical
or
Radical Expression
Principal square root
Positive square root of a positive real number
Square Root
Mistakes made with Square Roots and Radical Expressions
Square root of x�
Index
Square roots with expressions with even and odd exponents
Cube root
Types of roots
Complex number
i and i�
(Complex numbers)
Square root of a negative number
(Complex numbers)
Number Types
Equations with Complex Numbers
Addition & Subtraction with Complex Numbers
Common Error when multiplying Complex Numbers
Multiplication with Complex Numbers
1. FOIL
2. Simplify
3. Standard Form
Conjugate
(a+b) and (a-b) are called this
Division with Complex Numbers
1. Multiply both the numerator and denominator by the complex conjugate of the denominator.
2. Simplify the resulting products in both the numerator and denominator.
3. Write in standard form.
Powers of i
Only 4 answers
1
-1
i
-i
i� =
i? =
i? =
i�� =
1
i� =
i? =
i�? =
-1
i� =
i? =
i�� =
-i
i? =
i? =
i
Factoring two squares
(Example: x�+9)
(Factoring Complex Numbers)
This is "not factorable" using real factors, but can be factored with complex conjugates.
Solving Quadratic Equations using Square Root Property
Quadratic Formula
(used to solve ax�+bx+c=0)
Quadratic Formula common mistakes
Quadratic Formula, solving the discriminant
(b�-4ac)
Projectile Formula
h = height of object, in feet
t = time (seconds) the object is in the air
v? = beginning velocity, (feet per second)
h? = beginning height (0 if start from ground)
(This BETTER not be on the test)
Volume of a box formula
V=lwh
(Volume = Length x width x height)
Quadratic functions
y=ax�+bx+c
A function that can be written in the form above. Every graph of this function is a parabola.
Parabola
Graphs that have a curve
Example here is y=x�-4x+3
Vertex
the "turning point" of a parabola
Example here is (2,-1)
Line of Symmetry
(axis of symmetry)
the line that passes through the vertex and creates the mirror image
Example here is x=2
y=ax�
(items to remember when graphing)
domain: x|x is any real number
vertex: (0,0)
line of symmetry: x=0 (y-axis)
bigger |a|: narrower the opening
smaller |a|: wider the opening
if a>0, the parabola opens upward
if a<0, the parabola opens downward
y=ax�+k
(items to remember when graphing)
k creates a vertical shift
domain: x|x is any real number
vertex: (0,k)
line of symmetry: x=0 (y-axis)
bigger |a|: narrower the opening
smaller |a|: wider the opening
if a>0, the parabola opens upward
if a<0, the parabola opens downward
y=a(x-h)�
(items to remember when graphing)
h creates a horizontal shift
domain: x|x is any real number
vertex: (h,0)
line of symmetry: x=h
bigger |a|: narrower the opening
smaller |a|: wider the opening
if a>0, the parabola opens upward
if a<0, the parabola opens downward
y=a(x-h)� +k
(items to remember when graphing)
vertical and horizontal shifts
domain: x|x is any real number
vertex: (h,k)
line of symmetry: x=h
bigger |a|: narrower the opening
smaller |a|: wider the opening
if a>0, the parabola opens upward
if a<0, the parabola opens downward
y=ax�+bx+c
(items to remember when graphing)
Change into form y=a(x-h)�+k
once in this form, you can easily find the vertex, line of symmetry and range
Minimum and Maximum values
Exponents that are fractions
(Example 6 to the � power)
Exponents to the nth
Radical Notation
General Form of Fraction Exponents
and Radical Notation
Sum of two functions
Difference of two functions
Product of two functions
Quotient of two functions
Composite function
&
Domain of that function
One-to-one functions
(1-1 functions)
each value of y in the range must only have one corresponding value of x in the domain.
(x,y) meaning for each y, there are no duplicate x's
Horizontal line test
A function is a one-to-one if not horizontal line intersects the graph of a the function at more than one point.
Inverse functions
if � is a one-to-one function with (x,y)
the the inverse �?� is also one-to-one with (y,x)
To determine whether two functions are inverses
Finding the Inverse of a one-to-one function