Commutative Property of Addition
If a and b are real numbers,
then a + b= b+a
Example: 3+5=8 and 5+3=8
Commutative Property of Multiplication
If a and b are real numbers,
then ab = ba
Example: (-4)(6)=-24 and
(6)(-4)=-24
Associative Property of Multiplication
If a,b,and c are real numbers,
then (a+b)+c=a+(b+c)
Example: (-8�-2)�3 = 16�3=48 and
-8�(-2�3)=-8�(-6)=48
Distributive Property
If a,b, and c are real numbers,
then a(b+c) = ab + ac
Example: 3(x-4) = 3� x + 3 � (-4)
= 3x - 12
Exponential Notation
A method in mathematics that allows the representation of numbers in shorter form; how many times to use the number in a multiplication.
Simplifying Expressions
Reducing an expression by combining like terms and using order of operations.
Step #1 Simplifying Expressions: Identify and Combine Like terms
Identify: Like terms have the same configuration of variables, raised to the same powers. They must have the same variable or variables, or none at all, and each variable must be raised to the same power, or no power at all.
Example: 1 + 2x - 3 + 4x.
Comb
Step #2 Simplifying Expressions: Simplify the Expression
Construct an expression from your new, smaller set of terms. Get a simpler expression that has one term for each different set of variables and exponents in the original expression. This new expression is equal to the first.
New expression is 6x - 2. This
Step #3 Simplifying Expressions: Order of operations
Use the acronym PEMDAS to remember the order of operations.
-Parentheses
-Exponents
-Multiplication
-Division
-Addition
-Subtraction
Factoring: #1 Identify the greatest common factor
Identify the greatest common factor in the expression. Factoring is a way to simplify expressions by removing factors that are common across all the terms in the expression. To start, find the greatest common factor that all of the terms in the expression
Factoring #2: Dividing the terms by the GCF
Divide every term in your equation by the greatest common factor. The resulting terms will all have smaller coefficients than in the original expression.
Factor out 9x � + 27x -3 by its greatest common factor of 3. To do so divide each term by 3.
1. 9x� d
Factoring #3: Make new expression equal to the old one
To make our new expression equal to the old one, we'll need to account for the fact that it has been divided by the greatest common factor. Enclose your new expression in parentheses and set the greatest common factor of the original equation as a coeffic
Factoring to Simplify Fractions
Let's say our original example expression, 9x2 + 27x - 3, is the numerator of a larger fraction with 3 in the denominator. This fraction would look like this: (9x2 + 27x - 3) � 3. We can use factoring to simplify this fraction.
Let's substitute the factor
Use square factors to simplify radicals
Expressions under a square root sign are called radical expressions. Simplify by identifying square factors and perform the square root operation on these separately to remove them from under the square root sign.
Example: ?(90)
If we think of the number
Multiplying and Dividing Exponential Terms
Add exponents when multiplying two exponential terms; subtract when dividing. This concept can also be used to simplify variable expressions.
Example: 6x� � 8x? + (x�? - 15)
(6 � 8)� + 4 + (x �? - 15)
48x7 + x2
Zero Exponent Rule
ANY BASE RAISED TO THE ZERO POWER IS EQUAL TO 1
Exponents
Shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base to something simpler.
Exponents: Product with the same base
When multiplying like bases, keep the base the same and add the exponents.
Exponents: Quotient with the same base
When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.
Exponents: Quotients to a power
When raising a fraction to a power, distribute the power to each factor in the numerator and denominator of the fraction.
Exponents: Negative power
Negative exponents signify division. In particular, find the reciprocal of the base.
When a base is raised to a negative power, reciprocate (find the reciprocal of) the base, keep the exponent with the original base, and drop the negative.
Exponents: Power to a power
When raising a base with a power to another power, keep the base the same and multiply the exponents.
Exponents: Product to a power
When raising a product to a power, multiply the power to each factor.
Exponents: Zero Power
Anything raised to the zero power is one.
Exponents: Quotient with a Negative Power
Negative exponents signify division - find the reciprocal of the base. When a denominator is raised to a negative power, move the factor to the numerator, keep the exponent but drop the negative.
1/5?� (with five to the -3rd power) = 5�
radical expression
an algebraic expression that includes a root. The root may be a square root, a cube root, or any other power. Simplifying a radical expression can help you solve an equation. Simplifying radical expressions involves removing the root when possible, or red
Simplify any radical expressions that are perfect squares
A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. To simplify a radical expression that is a perfect square, simply remove the radical sign and write the number that is the square root o
Simplify any radical expressions that are perfect cubes
A perfect cube is the product of any number that is multiplied by itself twice, such as 27, which is the product of 3� 3 � 3. To simplify a radical expression when a perfect cube is under the cube root sign, simply remove the radical sign and write the nu
Break down an imperfect radical expression into its multiples
step 1: The multiples are the numbers that multiply to create a number -- for example, 5 and 4 are two multiples of the number 20. To break down an imperfect radical expression by its multiples, write down all of the multiples of that number (or as many a
Find a perfect square in the variable
The square root of a to the second power would be a. The square root of a to the third power is broken down into the square root of a squared times a -- this is because you add exponents when you multiply variables, so that a squared times a reverts back
Simplify a radical expression with variables and numerals that is a perfect square
EXAMPLE: ?36a�
take apart the expression by first looking for perfect squares in the numbers, and then looking for perfect squares in the variables. Then, remove the radical sign and let their square roots remain. the square root of 36 x a squared.
1. 36
Simplify a radical expression with variables and numerals that is not a perfect square
? 50a�
break down the expression into numerals and variables, and search for perfect squares within the multiples of both. Then, pull any perfect squares out of the radical expression.
-Break down 50 to find any multiples that are a perfect square. 25 � 2
Polynomial
The sum or difference of one or more monomials. A polynomial with two terms is a binomial. With three terms is a trinomial The degree of a polynomial is the degree of the highest monomial term. A polynomial can have constants, variables and exponents, but
Adding Polynomials
1. Place all like terms together. (all x's; all yx's)
2. Add all like terms.
Subtracting Polynomials
1. In the set that is being subtracted: Reverse the minus and plus signs within the parenthesis.
2. In column format perform the calculations.
3. Remove any solutions with "0
Multiplying Polynomials
First multiply the constants, then multiply each variable together and combine the result
Example: (2xy)(4y)
Equals: 2�4�xy�y
Equals: 8xy (and y is to the 2nd power)
Dividing Polynomials
Dividing polynomials using long division:
The numerator (top number) is divided by the denominator (bottom number.
-Divide the first term of the numerator by the first term of the denominator, and put that in the answer.
-Multiply the denominator by that
Factoring Polynomials
Factoring polynomial expressions is not the same but similar to factoring numbers. When factoring numbers or factoring polynomials, you are finding numbers or polynomials that divide out evenly from the original numbers or polynomials. But in the case of
Factoring Polynomials Example
Factor 3x - 12:
The only thing common between the two terms (that is, the only thing that can be divided out of each term and then moved up front) is a "3".
1. So factor this number out to the front:
3x - 12 = 3( )
2. When the "3" is divided out of the "3
Rational Expressions
An algebraic expression that can be written as a fraction whose numerator and denominator are polynomials and the denominator does not equal "0
Rational Expressions Example
Simplify the following expression:
(2x) � (x�)
1. Cancel off any common numerical or variable factors.
The numerator factors as (2)(x); the denominator factors as (x)(x). Anything divided by itself is just "1", so cross out any factors common to both the
Linear Expressions
A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a ? 0.
A linear expression is an expression with a variable in it; however, the variable is only raised to the first power.
Solving Linear Equations
Solve x + 6 = -3
Get the x by itself; that is, I want to get "x" on one side of the "equals" sign, and some number on the other side. I want just x on the one side, since the 6 is added to the x, you need to subtract ab 6 from the x in order to "undo" hav
Solving Linear Equations with Integer Coefficients
1. Simplify the equation by adding together all of the "x" values. For example, the equation x + 3x + 8 = 5� + 3 can be simplified by adding the x values on the left side of the equation: x + 3x = 4x. The equation becomes 4x + 8 = 5� +3.
2. Calculate all
Solving Linear Equations with Fractions
1. Multiply each side by the reciprocal of the fraction.
4a/5 = 12 The reciprocal of the fraction 4/5 = 5/4
5/4 numerator � 4a/5 = 5/4�12
2. Reduce and simplify.
a. 5/4 times 4/5 = 0 so you are left with "a" for the left side.
b. 5/4 times 12 = cross redu
Solving Linear Equations with Radicals
A radical expression is an equation in which at least one variable expression is stuck inside a radical, usually a square root. ?x
When you have a variable inside a square root, you undo the root by doing the opposite: squaring. For instance, given sqrt(x
Solving Linear Equations with Decimals
Decimal numbers are fractions in disguise, "clear the decimal" in equations with decimal numbers. Count the largest number of digits behind each decimal point and multiply both sides of the equation by 10 raised to the power of that number.
Example: 0.25
x and y intercepts
x-intercept is a point in the equation where the y-value is zero
y-intercept is a point in the equation where the x-value is zero
Whichever intercept you're looking for, the other variable gets set to zero.
Also: think of the following terms interchangeab
Finding x intercepts
Find the x- and y-intercepts of 25x� + 4y� = 9
"x-intercept(s):"
y = 0 for the x-intercept(s), so:
25x� + 4y� = 9
25x� + 4(0)� = 9
25x� + 0 = 9
x� = 9/25
x = � ( 3/5 )
Then the x-intercepts are the points (?3/5, 0) and (?�/?, 0)
Finding Y intercepts
Find the y-intercepts of 25x� + 4y� = 9
x = 0 for the y-intercept(s), so:
25x� + 4y� = 9
25(0)� + 4y� = 9
0 + 4y� = 9
y� = 9/4
=y � ( 3/2 )
Then the y-intercepts are the points
(0, 3/2) and (0, ?3/2)
Determine the domain and range of a function from its graph
Domain: All the x-values
Range: All the y-values
List all values without duplication
y= - 4? + 4
Linear Expressions: Converting verbal statements into mathematical statements
Read the question carefully and recognize the key words as:
1. Add:
-Sum; More than; increased by; combined, together; total of; added to
2. Subtract: Decreased by; minus, less; difference between/of; less than, fewer than
3. Multiply: Of; times, multipli
Word problem tips
1. Read the problem entirely before trying to solve anything. Get a good feeling of the problem and try to see what information is available and what is missing.
2. Organize the information by naming the problem. Use variables for the unknown.
3. Look for
Word Problem Example - Painting a Fence
Tina can paint a fence in 12 hours. Tony helps and they paint the fence in one hour. How fast can Tony paint the fence by himself?
X represents how fast Tony can paint the fence by himself: 1/X
Tina can paint the fence by herself in 12 hours. So she can c
Word Problem: Age
In February of the year 2000, Tina was one more than eleven times as old as my son Raymond. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000?
Here's how you'd figure out his age:
1. Name the items
Geometry Word Problems
1. You need to know the basics of geometric formulas:
Rectangles:
a. Area: length times the width
b. Perimeter means "length around the outside". 2�l + 2�w
Triangles:
a. Area: (1/2)bh
b. Perimeter: The sum of all three sides.
Circles:
a. Area: (?)p�
b. Ci
Coin word problems
TIPS:
1. Convert the relationships between the numbers of coins (if given) into equations
2. Convert the statements about the values of the coins (if given) into equations that state the values all in the same unit (for instance, in cents).
3. Label every
Distance Word Problems
Involve something travelling at some fixed and steady ("uniform") pace ("rate" or "speed"), or else moving at some average speed. Whenever you read a problem that involves "how fast", "how far", or "for how long", you should think of the distance equation
Number Word Problems
The sum of two consecutive integers is 15. Find the numbers.
What you know:
1. You are adding two numbers
2. The sum is 15
3. The second number is one more than the first (consecutive integers)
Represent the first number by "n" and the second number will
Twice the larger of two numbers
Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers?
The point of exercises like this is to give you practice in unwrapping and unwinding t
Percent of Word Problems
When you learned how to translate simple English statements into mathematical expressions, you learned that "of" can indicate "times". This frequently comes up when using percentages.
If you need to find 16% of 1400, you first convert the percentage "16%
Complex Numbers
A combination of a "real" number and an "imaginary" number.
Imaginary Numbers: When squared, they give a negative result.
We have to imagine that there is a number because we need it.
i=??1
A complex number is a number that can be expressed in the form a
Complex Numbers in Standard Form example
What is the standard form of (3 - ? -5)(2 + ?-10) ?
(3 - i?5)(2 + i?10)=(6 + ?50) + i(3?10 - 2?5)
Slope of a Line
- Measures the steepness every nonvertical line
- Positive slope: Line goes up from left to right
- Negative slope: Line goes down from left to right
- Zero slope: Horizontal Line
- Undefined slope: Vertical Line
Definition of Slope
If x? ? x? the slope of a nonvertical line passing through distinct points(x?y?) and (x? x?) is given by:
m=
rise/run = change in y/change in x = y?-y?/x?-x?
Point-slope Form of the Equation of a Line
Given the slope of a line M and a point of the line (x?, y1), the point-slope form of the equation of a line is given by:
y - y� = m(x -x?)
You must know the slope of the line a and a point on the line.
Slope-Intercept Form of the Equation of a Line
Given the slope of a line m and the y-intercept, b, the slope-intercept form of the equation of a line is given by: y=mx+b
Standard Form Equation of a Line
The standard form of an equation of a line is given by
Ax + By = C, where A, B, and C are real numbers such that A and B are both not zero.
This text always includes nonfractional coefficients and A is always greater than or equal to zero.
Eliminate fract
Horizontal and Vertical Lines
Any horizontal line that contains the point (a,b) the equation of that line is y = b and the slope is m = 0.
Any vertical line that contains the point (a,b) the equation of that line is x = a and the slope is undefined.
Given the point (-5,12)
The equatio
Definition of a Relation
A relation is a correspondence between two sets A and B such that each element of set a corresponds to one or more elements in set B
Domain: Elements in Set A
Range: Elements in Set B
Definition of a Function
A function is a relation such that for each element in the domain, there is exactly one corresponding element in the range.
There are no duplication of x
Determining Whether Equations Represent Functions
To determine whether an equation represents a function, we must show that for any value in the domain, there is exactly one corresponding value in the range.
To make this determination, solve for y. If the solution indicates there are two possible y-value
Function Notation
Instead of using the variable y, letters such as f, g, or h (and others) are commonly used for functions.
If we want to name a function f, then for any x-value in the domain, we call the y-value (or function value) f(x).
f(x) is read "f of x" or "the valu
Vertical Line Test
A graph in the Cartesian plane is the graph of a function if and only if no vertical line intersects the graph more than once.
Determining the Domain of a Function Given the Equation
Domain is the set of all values of x for which the function is defined.
A number x = a is in the domain of a function � if �(a) is a real number.
The domain of every polynomial function is all real numbers
Inverse Functions
1. Only one to one functions have inverses
If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other.
2. If f and g are inverses of each other then both are one to one functions.
3. f and g are inverses of each other i
whole number
0,1,2,3,4...
natural number
123456789
integer
A whole number (not a fraction) that can be positive, negative, or zero
rational numbers
Any number that can be written as a fraction using only integers.
irrational numbers
numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.
real numbers
the set of rational numbers and irrational numbers
absolute value
The distance of a number from zero on a number line
factoring trinomials into two binominals with a leading co-efficient of one
first look at the constant on the end, and you want to find two numbers that multiply to equal that constant but that also add up to be the middle co-efficient. These are going to be the numbers that go into your parentheses.
Example: x�+�?12
. The consta
factoring trinomials into two binominals without a leading co-efficient of one
Example: 2x� - 18x + 40
is there a greatest common factor among all of our terms, and we see that 2 is a greatest common factor.
Factoring out the 2 leaves you with:
2(x�-9x+20)
Now apply the two step process to factor what is inside the parenthesis.
- No
Interval
a set containing all real numbers or points between two given real numbers or points
Interval notation
A shorthand way of writing intervals using parentheses and brackets.
x+4<4x-5
. Subtract 4 from both sides:
x<4x-9
. Subtract 4x from both sides:
-3x<-9
.Multiply both sides by -1 (reverse the inequality):
(-1)(-3x)>(-2)(-9)
3x>9
Solution: x>3
Interval No