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

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 0.00000000123 = 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 -2 + -2 = -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

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.

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

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 0.00000000123 = 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

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

Example Problem 2x 3x + 4 6x2 – 7x – 20 2x 3x + 4 6x2 + 8x 1 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

II I III IV Steps for Factoring by Grouping 1 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