1. Keywords for Addition (+) Keywords for Subtraction (+)
What is the sum of 9 and y?
9 + y
What is four less than y?
y – 4
Express the number (x) of peaches increased by 3.
x + 3
What is nine less than a number (y)?
y – 9
Express the total weight of x and y in kilograms.
x + y
What if the number of trees was reduced by 8?
x – 8
What is the difference of my height (x) by your height (y)?
x – y
Keywords for Multiplication (* x)
Keywords for Division ( / )
What is y multiplied by 14?
13y
What is the quotient of y and 2?
y/2
Three swimmers averaged “y” minutes. Express their total swimming time. 3y
Three girls rent an apartment for $”x” a month. What will each have to pay?
x/3
I drive my car at 60 miles per hour. How far will I go in “x” hours?
60x
“y” items cost a total of $30.00 Express their average cost.
30/y

2. Type
Definition
Examples
Real numbers
All numbers that can be represented on a real number line
-2.26, 0, , 10
Rational number
All numbers that can be expressed as a quotient of 2 integers (denominator not 0)
3.25, 500
Irrational number
Non-terminating, non-repeating decimal numbers ,
Natural number
The numbers used for counting
1, 2, … infinity
Whole number
The numbers 0 and all natural numbers
0, 1, … infinity
Integer
0, negative integers, and positive integers
-4, 0, 150
Positive integer
Whole numbers greater than 0
1, 5, 100
Negative integer
Whole numbers less than 0
-1, -5, -100
Absolute value
The distance between the number and 0 on the number line
|1| = 1, |-3| = 3
If signs are
Product is
Like
Positive (+)
Unlike
Negative (-)
If signs are
Quotient is
Like
Positive (+)
Unlike
Negative (-)
Exponent Rule
Example
Any number raised to the zero power (except 0) equals 1.
n0=1
Any number raised to the power of one equals itself.
n1=n
To multiply terms with the same base, add the exponents.
(n3) (n5) = n8
To divide terms with the same base, subtract the exponents.
n5 ÷ n3 = n2
To raise a power to a power, multiply the exponents.
(x2) 3 = x6
When a product has an exponent, each factor is raised to that power.
(xy) 3 = x3y3
A number with a negative exponent equals its reciprocal with a positive exponent.
n-5 =
1,230,000,000 = 1.23 x 109
Long number
Scientific notation
Long number
= 1.23 x 10-9
Scientific notation
-2 – 2 = -4
2 + 2 = 4
When you subtract a positive number, move left
When you add a positive number, move right
= -4
2 – (-2) = 4
When you add a negative number, move left
When you subtract
a negative number, move right 2 -2 4 6 -4 -6 2 -2 4 6 -4 -6

3. Example Terms Like or Not? Reason Like
2 and 4
Like
Neither term has a variable. They are constants.
4x and 3
Not
The second term has no variable.
3y2 and 3y
The y variable of the second term has no exponent.
2x2y2 and 7x2y2
Both terms have the same variables and exponents.
2x2y2 and 7x2y5
Both terms have the same variables but not the exponents.

4. Commutative Properties
Addition
a + b = b + a
Multiplication
ab = ba
Addition Property of Equality
If a = b,
then a + c = b + c
for any real numbers a, b, and c.
Associative Properties
Addition
(a + b) + c = a + (b + c)
Multiplication
(a b)c = a(b c)
Addition Property of Multiplication
If a = b,
then a · c = b · c
for any real numbers a, b, and c.
Distributive Property
a (b + c) = ab + ac
Identity Properties
a + 0 = a
0 + a = a
a · 1 = a
1 · a = a
Inverse Properties
a + (- a) = 0
- a + a = 0
Inverse Properties
a ≠ 0
a · = 1
· a = 1

5. Exponent Rule
Example
Any number raised to the zero power (except 0) equals 1.
n0=1
Any number raised to the power of one equals itself.
n1=n
To multiply terms with the same base, add the exponents.
(n3) (n5) = n8
To divide terms with the same base, subtract the exponents.
n5 ÷ n3 = n2
To raise a power to a power, multiply the exponents.
(x2) 3 = x6
When a product has an exponent, each factor is raised to that power.
(xy) 3 = x3y3
A number with a negative exponent equals its reciprocal with a positive exponent.
n-5 =
1,230,000,000 = 1.23 x 109
Long number
Scientific notation
Long number
= 1.23 x 10-9
Scientific notation
Exponent
Constant
Polynomial Rules
Example
No division
2/(x+2)
No negative exponents
n-5 or
No fractional exponents
√x or 4x 1/2
Variable
1first
4last
3inside
2outside
No variables
Power of 2
Power of 1
Correct
You must divide the denominator by both terms
Incorrect
You cannot divide by one term only 1 1

6. Step 2 Step 1 Step 3 Step 4 Step 5 Step 6 Step 7
Divide each term by 2x
Simplify each term
Step 2
Step 1 2x
3x + 4
Divisor 2x
6x2 – 7x – 20
3x + 4
Quotient of
6x2 3x
6x2 – 7x – 20
6x2 + 8x
3x + 4
6x2 – 7x – 20
Product of 2x(3x+4)
Dividend
Step 3
Step 4
Step 5 2x 2x
2x – 5
Quotient of
-15x 3x
3x + 4
6x2 – 7x – 20
- (6x2 + 8x)
- 6x2 - 8x
Subtract
- 6x2 - 8x
0 – 15x
- 15x - 20
- 15x - 20
Step 6
Step 7
2x – 5
2x – 5
Answer
2x – 5
- 6x2 - 8x
- 6x2 - 8x
3x + 4
6x2 – 7x – 20
- 15x - 20
- 15x – 20
- 15x - 20
Product of
-5(3x+4)
-(- 15x – 20)
Subtract

7. Example Problem 2x 3x + 4 6x2 – 7x – 20 2x 3x + 4 6x2 + 8x1 1 2 3
5(7x2 – 5x + 1) = 35x2 – 25x + 5 2 3
Divide each term by 2x
Simplify each term
Example Problem
Quotient of
6x2 3x 2x
3x + 4
Divisor 2x
6x2 – 7x – 20
3x + 4
6x2 – 7x – 20
6x2 + 8x
3x + 4
Dividend
6x2 – 7x – 20
Product of 2x(3x+4)
Quotient of
-15x 3x 2x 2x
3x + 4
6x2 – 7x – 20
2x – 5
- (6x2 + 8x)
- 6x2 - 8x
Subtract
- 15x - 20
- 6x2 - 8x
0 – 15x
- 15x - 20
2x – 5
2x – 5
2x – 5
- 6x2 - 8x
- 6x2 - 8x
3x + 4
6x2 – 7x – 20
- 15x - 20
- 15x – 20
- 15x - 20
Product of
-5(3x+4)
-(- 15x – 20)
Subtract

8. II I III IV Steps for Factoring by Grouping1 1 2 3
(5c – 3)(2c + 5) = 10c2 + 25c – 6x + 15 2 3
Steps for Factoring
by Grouping
Yes
Factor the GCF from each of the 4 terms.
Are there any factors common to all four terms?
Arrange the terms so the 1st two have a common factor and the last two have a common factor.
Use the distributive property to factor each group two terms.
Factor the GCF from the results. No
y-axis
y-axis
(2,4) II
(1, 3) I
Quadrant
X-axis
Y-axis I + II ―
III IV 4 3
(-4,2) 3 2
(0,2) 2 1
Origin 1
x-axis -4 -3 -2 -1 1 2 3 4
x-axis -4 -3 -2 -1 1 2 3 4
III IV -1 -1 -2 -2
(-1, -3) -3
(4, -3) -3