# Section 1.9 Algebraic Expressions: The Distributive Property

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Section 1.9 Algebraic Expressions: The Distributive Property Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Apply the Distributive Property. The Associative Properties (1.1) The Commutative Properties (1.1) Multiplying real numbers (1.5) INTRODUCTION In Section 1.8, we added and subtracted terms, and we now turn our attention to multiplication. In this section we learn how to multiply a term by an integer, a skill that will prove helpful in the coming chapters. We also take another look at the Distributive Property, this time with variables. MULTIPLYING A TERM BY AN INTEGER There is occasion when we must multiply a term by an integer, such as 2 (7x) or -5 (3y 2 ). Because multiplication is the only operation present, we can apply the Associative Property and regroup the multiplication. For example, 2 (7x) -5 (3y 2 ) = (2 7)x = (-5 3)y 2 = 14x = -15y 2 Once we understand this procedure, it is common to simplify these expressions by multiplying the numbers directly: 2(7x) = 14x and -5 (3y 2 ) = -15y 2. Note two things: 1. the multiplication symbol (raised dot) is not required in the written product, and 2. the variable factor is unaffected by the multiplication of the two numbers. It s also possible that the integer is on the right side, as in this expression, (-9k)(-4). Again, we can multiply the numbers directly, as shown at right: (-9k) (-4) = 36k Algebraic Expressions: The Distributive Property page

2 Example 1: Simplify each expression by multiplying. a) 4(5m) b) -3(8p) c) -6(-w 2 ) d) (y)(-7) Procedure: Multiply the numbers directly; the variable factors are unaffected. Also, in part c), the coefficient of w 2 is -1, and in part d), the coefficient of y is 1. Answer: a) 20m b) -24p c) 6w 2 d) -7y You Try It 1 Simplify each expression by multiplying. Use Example 1 as a guide. a) 9(4c) b) -6(5y 2 ) c) -1(-12a) d) -6(v) e) (-r)(-4) f) (-2b 2 )6 THE DISTRIBUTIVE PROPERTY Recall, from Section 1.1, the Distributive Property of Multiplication over Addition. The number that is distributed is called the multiplier. The multiplier can be on the right side or the left side of the parentheses. The Distributive Property of Multiplication over Addition 1. The multiplier on the left side: 2. The multiplier on the right side: b(c + d) = bc + bd (c + d)b = cb + db Note: The Distributive Property is often written without showing the multiplication sign. With the multiplication sign, the distributive Property is b (c + d) = b c + b d. Algebraic Expressions: The Distributive Property page

3 Let s see why the distributive property works the way it does. Consider the example 3(x + 6). Because multiplication is an abbreviation for repeated addition, this means The sum of three (x + 6) s. This can be written as = (x + 6) + (x + 6) + (x + 6). The associative property can be used to write that expression without parentheses: = x x x + 6 We can then use the commutative property to reorganize the numbers: = x + x + x Let s group the x s separately from the 6 s: = (x + x + x) + ( ) Each grouping has three of something: = three x s + three 6 s Abbreviating each, we get: = 3(x) + 3(6) So, from the beginning: 3(x + 6) = 3x + 3(6) Behold, the distributive property. = 3x + 18 Example 2: For each expression, apply the distributive property. a) 5(7y + 3) b) (-3x 2 + 9x)2 Procedure: Distribute the multiplier to each term in the quantity, then multiply the terms. Answer: a) 5(7y + 3) The multiplier is 5. Distribute 5 times to each term in the quantity. = 5(7y) + 5(3) Multiply the terms. = 35y + 15 b) (-3x 2 + 9x)2 The multiplier is 2. Distribute times 2 to each term in the quantity. = (-3x 2 )2 + (9x)2 Multiply the terms. = -6x x Algebraic Expressions: The Distributive Property page

4 You Try It 2 For each expression, apply the distributive property. Use Example 2 as a guide. a) 6(4x + 3) b) (-6m 2 + 8)5 c) 4(2y 4 + 6y 2 ) Technically, the distributive property is the product of a number (the multiplier) and a quantity (a sum of two or more terms). However, because we are familiar with negative numbers and how to multiply with them, we can slightly modify the distributive property to include subtraction. For example, if we wish to distribute the multiplier 4 through the quantity (5x 3), the written product looks like this: 4(5x 3). We can, however, think of the quantity 5x 3 as 5x + (-3), effectively changing the product to 4 (5x +(-3)). Let s look at it one step at a time. 4(5x 3) Subtraction can be rewritten as a sum: adding the opposite. = 4(5x + (-3)) We can distribute the multiplier through to each term in the sum. = 4(5x) + 4(-3) Multiply: 4 (-3) = -12 = 20x + (-12) Rewrite the terms in a more simplified form; change plus (-12) to minus 12 = 20x 12 Actually, we don t need to go through all of that work if we remember that the sign in front of a term belongs to that term. For example, in the quantity (5x 3), the terms are 5x and -3. So, in effect, when distributing the multiplier, 4, through to the sum (5x 3) we are multiplying 4 by both +5x and -3. As we multiply 4 by (+5x) we get +20x; and as we multiply 4 by (-3) we get -12. The +20x shows up as just 20x, but the -12 shows up as minus 12. In other words, 20x (5x 3) Treat 5x as +5x and minus 3 as (5x! 3) = 20x 12 4(+5x) = +20x; 4(-3) = -12, or minus 12. Notice that this requires fewer steps than before. This also means that we can distribute over subtraction as long as we understand that subtraction means the second term is negative. Algebraic Expressions: The Distributive Property page

5 Example 3: For each expression, apply the distributive property. a) 9(4y 3 5) b) 2(4 6c) c) (5w 2 2)8 Procedure: Identify the multiplier and the signs of the terms in the quantity. Then, distributive and multiply in one step. Answer: a) 9(4y 3 5) The multiplier is 9. The terms in the quantity are 4y 3 and -5. = 36y 3 45 b) 2(4 6c) The multiplier is 2. The terms in the quantity are 4 and -6c. = 8 12c c) (5w 2 2)8 The multiplier is 8. The terms in the quantity are 5w 2 and -2. = 40w 2 16 You Try It 3 For each expression, apply the distributive property. Use Example 3 as a guide. a) 3(8x 9v) b) (-7x 2 4)3 c) 2(2m 2 5k) THE DISTRIBUTIVE PROPERTY WITH A NEGATIVE MULTIPLIER We can use the Distributive Property even when the multiplier is negative. We must be more careful when multiplying each term by a negative number, and we must continually remember that the sign in front of a number belongs to that number. Consider, for example, the product of the multiplier -2 and the sum (4x 3). As a product, this looks like -2(4x 3): - 8x + 6-2(4x 3) Treat 4x as +4x and 3 as (+4x) = -8x; -2(-3) = +6, or plus (4x 3) = -8x + 6 Algebraic Expressions: The Distributive Property page

6 Let s practice using the distributive property: Example 4: For each expression, apply the distributive property. a) -5(7x + 3) b) -2(4c 6) c) -8(-5 + 2y) d) -3(-6 y) Procedure: Identify the multiplier and the signs of the terms in the quantity. Then, distributive and multiply in one step. Answer: a) -5(7x + 3) -5 (+7x) = -35x; -5(+3) = -15, which shows up as minus 15. = -35x 15 b) -2(4c 6) -2 (+4c) = -8c; -2(-6) = +12, which shows up as plus 12. = -8c + 12 c) -8(-5 + 2y) -8 (-5) = +40; -8 (+2y) = -16y, which shows up as minus 16y. = 40 16y d) -3(-6 y) -3 (-6) = +18; -3 (-y) = +3y, which shows up as plus 3y. = y You Try It 4 For each expression, apply the distributive property. Use Example 4 as a guide. a) -5(6x + 3) b) -2(8k 3 10) c) -7(-4w 2 2w) Algebraic Expressions: The Distributive Property page

7 You Try It Answers You Try It 1: a) 36c b) -30y 2 c) 12a d) -6v e) 4r f) -12b 2 You Try It 2: a) 24x + 18 b) -30m c) 8y y 2 You Try It 3: a) 24x 27v b) -21x 2 12 c) 4m 2 10k You Try It 4: a) -30x 15 b) -16k c) 28w w Section 1.9 Exercises Think Again. The distributive property can be used to multiply 8 16 by treating 16 as (10 + 6): 8 (10 + 6) = = = 128 We could also use subtraction and treat 16 as (20 4): 8 (20 4) = = = Show how the distributive property can be used with subtraction to multiply Show how the distributive property can be used with subtraction to multiply Show how the distributive property can be used with subtraction to multiply Show how the distributive property can be used with subtraction to multiply Focus Exercises. Simplify each expression by multiplying. 5. 8(5w) 6. 4(3x 2 ) 7. -2(10m) 8. -6(6k) 9. -7(-4r) (-9y 2 ) 11. (-5x 2 )9 12. (-2h)3 13. (5x)(-7) 14. (-4d)(-4) 15. (p)(-9) 16. (d 3 )(-4) (-x) 18. (3w 2 )(-1) 19. (-4y)(1) 20. (-2x)(-1) Algebraic Expressions: The Distributive Property page

8 For each expression, apply the distributive property (8x + 4) 22. 7(3m + 1) 23. 4(7p 2 + 3) 24. 9(3x 2 + 4y) 25. (5w + 8)3 26. (1 + 4v) (8r p)2 28. (a 2 + 3b) (7x 8) 30. 6(9m 2 5) 31. 8(1 2v) 32. (7 8x 2 ) (-6p + 1) 34. 2(-7w 2 + 5) 35. 1(-4 + 9q) 36. 1( x) 37. 4(-2x 10) 38. 3(-2q 2 8) 39. 9(-6 4h) (-6 3k 2 ) (5x 2 + 2x) (y 2 + 4y) (2w w 2 ) (8p 9p 2 ) (-3w 2 7y) (-5c 2 11d) (7a 2 9b) (4p 3 8q) (x + 4) (3y 10) (a + 1) (m + 1) (y 3 7y) (9m 4 6m) (-5k 3 6k) (-p 2 + 3p) (7m 3 + m) (9v 3 + 6v) (-x 5 3x 3 ) (-y 6 2y 4 ) Think Outside the Box. Multiply the terms. Use the meaning of the exponent to expand each, as necessary, and then multiply w(6w) x(3x 2 ) 63. 3m 3 (-10m) y 2 (-5y 3 ) Apply the Distributive Property to each expression x(3x 7) v(1 2v) 67. 6m 2 (3m 1) x 2 (-2x 2 + 5x) Algebraic Expressions: The Distributive Property page

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### 2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

### Chapter 13: Polynomials

Chapter 13: Polynomials We will not cover all there is to know about polynomials for the math competency exam. We will go over the addition, subtraction, and multiplication of polynomials. We will not

### Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 5 Subtracting Integers

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 5 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Using the Properties in Computation. a) 347 35 65 b) 3 435 c) 6 28 4 28

(1-) Chapter 1 Real Numbers and Their Properties In this section 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS The properties of the real numbers can be helpful when we are doing computations. In this

### Unit 3 Polynomials Study Guide

Unit Polynomials Study Guide 7-5 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial

### The Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,

1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition

### 1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

### HFCC Math Lab Intermediate Algebra - 7 FINDING THE LOWEST COMMON DENOMINATOR (LCD)

HFCC Math Lab Intermediate Algebra - 7 FINDING THE LOWEST COMMON DENOMINATOR (LCD) Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example

### Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write

4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### (- 7) + 4 = (-9) = - 3 (- 3) + 7 = ( -3) = 2

WORKING WITH INTEGERS: 1. Adding Rules: Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = - 9 The sum of a negative and a positive number: First subtract: The answer

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### Polynomial Expression

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### 2.3 Solving Equations Containing Fractions and Decimals

2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

### Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One

### Addition and Subtraction of Integers

Addition and Subtraction of Integers Integers are the negative numbers, zero, and positive numbers Addition of integers An integer can be represented or graphed on a number line by an arrow. An arrow pointing

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### 1.4 Variable Expressions

1.4 Variable Expressions Now that we can properly deal with all of our numbers and numbering systems, we need to turn our attention to actual algebra. Algebra consists of dealing with unknown values. These

### Equations, Inequalities, Solving. and Problem AN APPLICATION

Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a \$0 setup fee plus \$ per person. The cost of Hotel

### SIMPLIFYING ALGEBRAIC FRACTIONS

Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### 6.1 The Greatest Common Factor; Factoring by Grouping

386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### Factoring Trinomials of the Form x 2 bx c

4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

### Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

### Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A.

Order of Operations Section 1: Introduction You know from previous courses that if two quantities are added, it does not make a difference which quantity is added to which. For example, 5 + 6 = 6 + 5.

### Order of Operations More Essential Practice

Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure

### The Greatest Common Factor; Factoring by Grouping

296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### Monomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ.

Chapter 7.1 Introduction to Polynomials A monomial is an expression that is a number, a variable or the product of a number and one or more variables with nonnegative exponents. Monomials that are real

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### Section 4.3 Multiplying Polynomials

Section 4.3 Multiplying Polynomials So far, you have been introduced to the variety of definitions related to polynomials. You have also found the sum (addition) and difference (subtraction) between two

### Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

### SOLVING EQUATIONS BY COMPLETING THE SQUARE

SOLVING EQUATIONS BY COMPLETING THE SQUARE s There are several ways to solve an equation by completing the square. Two methods begin by dividing the equation by a number that will change the "lead coefficient"

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### Addition and Multiplication of Polynomials

LESSON 0 addition and multiplication of polynomials LESSON 0 Addition and Multiplication of Polynomials Base 0 and Base - Recall the factors of each of the pieces in base 0. The unit block (green) is x.

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### (2 4 + 9)+( 7 4) + 4 + 2

5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future

### Accentuate the Negative: Homework Examples from ACE

Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),

### Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

### 2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form.

Grade 9 Mathematics Unit : Powers and Exponent Rules Sec.1 What is a Power 5 is the BASE 5 is the EXPONENT The entire 5 is called a POWER. 5 = written as repeated multiplication. 5 = 3 written in standard

### Corinne: I m thinking of a number between 220 and 20. What s my number? Benjamin: Is it 25?

Walk the Line Adding Integers, Part I Learning Goals In this lesson, you will: Model the addition of integers on a number line. Develop a rule for adding integers. Corinne: I m thinking of a number between

### Algebra I Teacher Notes Expressions, Equations, and Formulas Review

Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts

### Adding and Subtracting Positive and Negative Numbers

Adding and Subtracting Positive and Negative Numbers Absolute Value For any real number, the distance from zero on the number line is the absolute value of the number. The absolute value of any real number

### Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

### To Evaluate an Algebraic Expression

1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum

### LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE

Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### Rational Numbers. Say Thanks to the Authors Click (No sign in required)

Rational Numbers Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org

### Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

### Fractions and Linear Equations

Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps

### Module 2: Working with Fractions and Mixed Numbers. 2.1 Review of Fractions. 1. Understand Fractions on a Number Line

Module : Working with Fractions and Mixed Numbers.1 Review of Fractions 1. Understand Fractions on a Number Line Fractions are used to represent quantities between the whole numbers on a number line. A

### Pre-Algebra - Order of Operations

0.3 Pre-Algebra - Order of Operations Objective: Evaluate expressions using the order of operations, including the use of absolute value. When simplifying expressions it is important that we simplify them

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Pre-Calculus II Factoring and Operations on Polynomials

Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...

### Solving Logarithmic Equations

Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### SAT Math Facts & Formulas Review Quiz

Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions

### PURPOSE: To practice adding and subtracting integers with number lines and algebra tiles (charge method). SOL: 7.3 NUMBER LINES

Name: Date: Block: PURPOSE: To practice adding and subtracting integers with number lines and algebra tiles (charge method). SOL: 7.3 Examples: NUMBER LINES Use the below number lines to model the given

### Equations Involving Fractions

. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation

### MULTIPLICATION OF FRACTIONS AND MIXED NUMBERS. 1 of 6 objects, you make 2 (the denominator)

Tallahassee Community College 0 MULTIPLICATION OF FRACTIONS AND MIXED NUMBERS You know that of is. When you get of objects, you make (the denominator) equal groups of the objects and you take (the numerator)

### P.E.R.T. Math Study Guide

A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT - A Math Study Guide 1. Linear Equations

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### Direct Translation is the process of translating English words and phrases into numbers, mathematical symbols, expressions, and equations.

Section 1 Mathematics has a language all its own. In order to be able to solve many types of word problems, we need to be able to translate the English Language into Math Language. is the process of translating

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### 7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1

7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

### Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

### Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in

### 1.4 Compound Inequalities

Section 1.4 Compound Inequalities 53 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities

### Accuplacer Arithmetic Study Guide

Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### Rational Expressions and Rational Equations

Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple

### Section 10.4 Vectors

Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such

### Math 25 Activity 6: Factoring Advanced

Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult

8. Radicals - Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens

### MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

### Clifton High School Mathematics Summer Workbook Algebra 1

1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:

### Training Manual. Pre-Employment Math. Version 1.1

Training Manual Pre-Employment Math Version 1.1 Created April 2012 1 Table of Contents Item # Training Topic Page # 1. Operations with Whole Numbers... 3 2. Operations with Decimal Numbers... 4 3. Operations

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### Boolean Algebra Part 1

Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems

### 1.3 Order of Operations

1.3 Order of Operations As it turns out, there are more than just 4 basic operations. There are five. The fifth basic operation is that of repeated multiplication. We call these exponents. There is a bit