# Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers

Size: px
Start display at page:

Transcription

1 Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Ex: 5 is a prime factor of 50. 1

2 Prime Factorization: the number written as a product of its prime factors. Ex: The prime factorization of 50 is We can represent prime factorizations two ways: factor tree repeated division by prime factors In both cases, the result is: 2 x 2 x 3 x 5 = 2 2 x 3 x 5. Use the method you prefer! 2

3 Divisibility RULES: A number is divisible by 2 if it is an even number 3 if the sum of the digits are divisible by 3 4 if the last two digits are divisible by 4 5 if it ends in 0 or 5 6 if both even and the sum of digits are divisible by 3 8 if the last 3 numbers are divisible by 8 9 if the sum of the digits are divisible by 9 Write the prime factorization of following: a) 45 b) 36 c) 110 d) 85 3

4 CYU pg. 135 Write the prime factorization of Page 140 #'s 4-6 Greatest Common Factor (GCF): the greatest number that divides into each number in a set of numbers. Ex: 5 is the GCF of 5, 10, and 15. You can use 2 methods to determine the GCF: 1. Rainbow method 2. Prime Factorization take the common factors between the set and multiply them together 4

5 Example 2 pg. 136 Determine the greatest common factor of 138 and 198. CYU pg. 136 Determine the greatest common factor of 126 and 144. Page 140 #'s 8a,c,e, 9a,b 5

6 Least Common Multiple (LCM): the least common multiple for a set of numbers. There are 2 ways to do this as well: 1. List the multiples of each number and pick the least common between the set 2. Use prime Factorization take the greatest of each number's power and multiply them all together Example 3 pg. 137 Determine the least common multiple of 18, 20, and 30. 6

7 CYU pg. 137 Determine the least common multiple of 28, 42, and 63. Page 140 #'s 10a,c,e, 11a,b?? 7

8 Section 3.2: Perfect Squares, Perfect Cubes, and Their Roots. Any whole number that can be represented as the area of a square with a whole number is a perfect square. The side length of the square is the square root of the area of the square. We write: 25 is a perfect square and 5 is its square root. 8

9 9

10 We can use prime factorization to determine if a number is a perfect square. If prime factors can be grouped into 2 equal groups, the number is a perfect square. Otherwise, the number is not a perfect square. Ex: Are the following numbers perfect squares? a) 6724 b) 1944 Example 1 pg. 144 Determine the square root of

11 CYU pg. 144 Determine the square root of Page 146 # 4 Any whole number that can be represented as the volume of a cube with a whole number edge length is a perfect cube. The edge length of the cube is the cube root of the volume of the cube. We write: 216 is a perfect cube and 6 is its cube root. 11

12 We can use prime factorization to determine if a number is a perfect cube. If prime factors can be grouped into 3 equal groups, the number is a perfect cube. Otherwise, the number is not a perfect cube. Ex: Are the following numbers perfect cubes? a) b)

13 = = = = = = = = = = = 3 (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) x (3 x 3 x 3) Example 2 pg. 145 Determine the cube root of

14 CYU pg. 145 Determine the cube root of Page 146 #'s 5 & 6 Example 3 pg. 146 A cube has a volume of 4913 cubic inches. What is the surface area of the cube? 14

15 CYU pg. 146 A cube has a volume 12,167 cubic feet. What is the surface area of the cube? Page 147 #'s 7-10 Section 3.3: Common Factors of a Polynomial What is the common factor in each of the following: a) 6 and 9 b) 5 and 15 c) 12 and 16 Hint: so the GCF of the numbers but pick the lowest exponent of the common variables d) 2x and 6x e) 8x and 16x f) 7x 2 y and 14xy 4 15

16 Factoring and expanding are inverse processes. After factoring, we can check by expanding. 16

17 Example 1 pg. 152 Factor each binomial. a) 6n + 9 b) 6c + 4c 2 17

18 CYU pg. 152 Factor each binomial a) 3g + 6 b) 8d + 12d 2 Page 155 #'s 7 & 8 18

19 Example 2 pg. 153 Factor the trinomial 5 10z 5z 2. Verify that the factors are correct (check by expanding). 19

20 CYU pg. 153 Factor the trinomial 6 12z 18z 2. Verify that the factors are correct (check by expanding). Page 155 #'s 9 & 10 Example 3 pg. 154 Factor the trinomial 12x 3 y 20xy 2 16x 2 y 2. Verify that the factors are correct (check by expanding). 20

21 CYU pg. 154 Factor the trinomial 20c 4 d 30c 3 d 2 25cd. Verify that the factors are correct (check by expanding). Page 156 #'s 15 & 16 21

22 22

23 Activate Prior Learning: Modelling Polynomials Write the polynomial represented by this set of algebra tiles. not a rectangle! 23

24 State the multiplication sentence for the following 24

25 Write a multiplication sentence with the product for each Can you make rectangles from these polynomials? If so, what are the factors of each. A. x 2 + 2x + 1 Note: These are examples with all positive terms B. y 2 + 3y + 2 C. r 2 + 7r + 10 D. w 2 + 7w

26 Working with negative terms. Write a multiplication sentence for each. Key: Blue = + White = Section 3.5: Polynomials of the Form x 2 + bx + c Polynomial of degree 2. If b, c are not 0, then there are 3 terms > trinomial x 2 + bx + c = 1x 2 + bx + c Since the leading coefficient is 1, this is a short trinomial. b is the coefficient of the second term. c is the constant term. x is the variable/unknown. 26

27 Generally, we like polynomials to be written in descending order. That is, the term with the largest degree first and the term with the smallest degree last. If we are given a polynomial is another order, we should rewrite it in descending order before proceeding. The variable/unknown is not always x. Ex: a 2 + 7a 18 z 2 12z t 2 16t Some variables aren't great choices: b, i, l, o, q, s Why? Multiplying Polynomials When multiplying polynomials, use the distributive property. Distributive Property: the property stating that a product can be written as a sum or difference of two products. Ex: a(b + c) = ab + ac Ex: (a+b)(c + d) = ac + ad + bc + bd After expanding with the distributive property, we simplify by combining like terms. Finally, we ensure polynomial is written in descending order. 27

28 Example 1 pg. 161 Expand and simplify. a) (x 4)(x + 2) b) (8 k)(3 k) CYU pg. 161 A) (c + 3)(c 7) B) ( 5 y)( 9 y) Page #'s 5, 9, 12, & 13 28

29 29

30 Are the following equal? (t 4)(t + 8) (t + 4)(t 8) Factoring a Short Trinomial To determine the factors of a short trinomial (x 2 + bx + c), determine two integers whose product is c and whose sum is b. limited possibilities Always start with the product as there are a infinite possibilities limited number of options. These integers are the constant terms in two binomial factors, each of which has x as its first term. 30

31 Example 2 pg. 163 Factor each trinomial a) x 2 2x 8 b) z 2 12z + 35 The order in which binomial factors are written does not matter. This is known as the commutative property. 31

32 CYU pg. 163 A) x 2 8x + 7 B) a 2 + 7a 18 Hint: creating a list of factors for c helps to determine n 1 and n 2. Page #'s 7, 11, & 14 32

33 Example 3 pg. 164 Factor 24 5d + d 2 33

34 CYU pg. 164 Page 167 #'s 15 & 17 34

35 Sometimes, the leading coefficient is not 1. However, if it is the GCF of all 3 terms, it can be factored out. However, it should tag along throughout the problem. Example 4 pg. 165 Factor 4t 2 16t CYU Pg. 165 Page 167 #'s 19 & 21 35

36 Short trinomials can also be factored using algebra tiles. 36

### Math 10-C Polynomials Concept Sheets

Math 10-C Polynomials Concept Sheets Concept 1: Polynomial Intro & Review A polynomial is a mathematical expression with one or more terms in which the exponents are whole numbers and the coefficients

### Unit 3 Factors & Products

1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors

### Review Unit Multiple Choice Identify the choice that best completes the statement or answers the question.

Review Unit 3 1201 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following numbers is not both a perfect square and a perfect cube? a. 531

### Multiplication of Polynomials

Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is

### Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +

### A-2. Polynomials and Factoring. Section A-2 1

A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring

### Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

### Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

### Chapter 5: Exponents and Polynomials

Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5

### Divisibility, Factors, and Multiples

Divisibility, Factors, and Multiples An Integer is said to have divisibility with another non-zero Integer if it can divide into the number and have a remainder of zero. Remember: Zero divided by any number

### Maintaining Mathematical Proficiency

Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h

### { independent variable some property or restriction about independent variable } where the vertical line is read such that.

Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with

### Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring

### Section 6.5 A General Factoring Strategy

Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b

### Chetek-Weyerhaeuser High School

Chetek-Weyerhaeuser High School Unit 1 Variables and Expressions Math RtI Units and s Math RtI A s 1. I can use mathematical properties to evaluate expressions. I can use mathematical properties to evaluate

### Review Notes - Solving Quadratic Equations

Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic

### Math Lecture 18 Notes

Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,

### Divisibility Rules Algebra 9.0

Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross

### MathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017

Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Unit 7: Factoring Quadratic Polynomials

Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples

### Algebra I. Exponents and Polynomials. Name

Algebra I Exponents and Polynomials Name 1 2 UNIT SELF-TEST QUESTIONS The Unit Organizer #6 2 LAST UNIT /Experience NAME 4 BIGGER PICTURE DATE Operations with Numbers and Variables 1 CURRENT CURRENT UNIT

### Algebra I Unit Report Summary

Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02

Adding and Subtracting Polynomials When you add polynomials, simply combine all like terms. When subtracting polynomials, do not forget to use parentheses when needed! Recall the distributive property:

### Expressions that always have the same value. The Identity Property of Addition states that For any value a; a + 0 = a so = 3

Name Key Words/Topic 2.1 Identity and Zero Properties Topic 2 Guided Notes Equivalent Expressions Identity Property of Addition Identity Property of Multiplication Zero Property of Multiplication The sum

### Can there be more than one correct factorization of a polynomial? There can be depending on the sign: -2x 3 + 4x 2 6x can factor to either

MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring

### LESSON 9.1 ROOTS AND RADICALS

LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical

### Algebra 1 Unit 6B Factoring

Algebra 1 Unit 6B Factoring Monday Tuesday Wednesday Thursday Friday 9 A Day 10 B Day 11 A Day 12 B Day 13 A Day Test Exponents and Polynomials Factor GCF and Trinomials box method Factoring Trinomials

Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic

### 6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property

6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.

Adding and Subtracting Polynomials Polynomial A monomial or sum of monomials. Binomials and Trinomial are also polynomials. Binomials are sum of two monomials Trinomials are sum of three monomials Degree

### Unit 3: Review for Final Exam

Unit 3: Review for Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Write the prime factorization of 1386. a. b. c. d. 2. Determine the greatest

### Florida Math Curriculum (433 topics)

Florida Math 0028 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular

### Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

### MA094 Part 2 - Beginning Algebra Summary

MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

### Part 2 - Beginning Algebra Summary

Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Order of Operations Expression Variable Coefficient

### Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

### Chapter 8 Class Notes 8-A1 (Lessons 8-1&8-2) Monomials and Factoring p Prime Factorization: a whole number expressed as the of factors.

Chapter 8 Class Notes Alg. 1H 8-A1 (Lessons 8-1&8-) Monomials and Factoring p. 40-4 Prime Factorization: a whole number epressed as the of factors. Tree Method: Ladder Method: Factored Form of a Monomial:

### MATH Spring 2010 Topics per Section

MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line

### KEY CONCEPTS. Factoring is the opposite of expanding.

KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add

### Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.

MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world

### INTRODUCTION TO FRACTIONS

INTRODUCTION TO FRACTIONS MEANING AND PROPERTIES OF FRACTIONS Fractions are used to represent parts of a whole. Example: What is the fraction of the shaded area? one-half one-quarter three-eighths 4 The

### Algebra 2. Factoring Polynomials

Algebra 2 Factoring Polynomials Algebra 2 Bell Ringer Martin-Gay, Developmental Mathematics 2 Algebra 2 Bell Ringer Answer: A Martin-Gay, Developmental Mathematics 3 Daily Learning Target (DLT) Tuesday

### MAFS Algebra 1. Polynomials. Day 15 - Student Packet

MAFS Algebra 1 Polynomials Day 15 - Student Packet Day 15: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent

Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Intensive Math-Algebra I Mini-Lesson MA.912.A.4.3

Intensive Math-Algebra I Mini-Lesson M912.4.3 Summer 2013 Factoring Polynomials Student Packet Day 15 Name: Date: Benchmark M912.4.3 Factor polynomials expressions This benchmark will be assessed using

### LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

### Chapter R - Review of Basic Algebraic Concepts (26 topics, no due date)

Course Name: Math 00023 Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 08/15/2014 End: 08/15/2015 Course Content: 245 topics Textbook: Miller/O'Neill/Hyde:

### A2T. Rational Expressions/Equations. Name: Teacher: Pd:

AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest

### Fantastic Factoring. Difference of Cubes. Difference of Squares. Sum of Cubes. Binomial Squares. Factor the following expressions

Fantastic Factoring Following are some factoring patterns that you might already recognize. x and y can both represent variables in the expressions, or y might be a constant. These rules work for all real

### Algebra I Polynomials

Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

### 27 Wyner Math 2 Spring 2019

27 Wyner Math 2 Spring 2019 CHAPTER SIX: POLYNOMIALS Review January 25 Test February 8 Thorough understanding and fluency of the concepts and methods in this chapter is a cornerstone to success in the

### Day 7: Polynomials MAFS.912.A-SSE.1.2, MAFS.912.A-SSE.2.3a,b, MAFS.912.A-APR.2.3, MAFS.912.F-IF.3.7c

Day 7: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent forms using factoring techniques use equivalent

### Ready To Go On? Skills Intervention 7-1 Integer Exponents

7A Evaluating Expressions with Zero and Negative Exponents Zero Exponent: Any nonzero number raised to the zero power is. 4 0 Ready To Go On? Skills Intervention 7-1 Integer Exponents Negative Exponent:

### Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

### Check boxes of Edited Copy of Sp Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and

Check boxes of Edited Copy of 10021 Sp 11 152 Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and Additional Topics Appendix Course Readiness Multiplication

### MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### My Math Plan Assessment #1 Study Guide

My Math Plan Assessment #1 Study Guide 1. Find the x-intercept and the y-intercept of the linear equation. 8x y = 4. Use factoring to solve the quadratic equation. x + 9x + 1 = 17. Find the difference.

Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........

### Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms

Polynomials Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms Polynomials A polynomial looks like this: Term A number, a variable, or the

### CHAPTER 1 POLYNOMIALS

1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6)

### review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

### Additional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping

Additional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping Find the greatest common factor of each list of monomials. 1. 10x and 15 x 1.. 3 1y and 8y. 3. 16 a 3 a, 4 and 4 3a

### Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

### Properties of Real Numbers

Pre-Algebra Properties of Real Numbers Identity Properties Addition: Multiplication: Commutative Properties Addition: Multiplication: Associative Properties Inverse Properties Distributive Properties Properties

### Factoring Polynomials. Review and extend factoring skills. LEARN ABOUT the Math. Mai claims that, for any natural number n, the function

Factoring Polynomials GOAL Review and extend factoring skills. LEARN ABOUT the Math Mai claims that, for any natural number n, the function f (n) 5 n 3 1 3n 2 1 2n 1 6 always generates values that are

### UNIT 2 FACTORING. M2 Ch 11 all

UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary

### Mathwithsheppard.weebly.com

Unit #: Powers and Polynomials Unit Outline: Date Lesson Title Assignment Completed.1 Introduction to Algebra. Discovering the Exponent Laws Part 1. Discovering the Exponent Laws Part. Multiplying and

### Prime Factorization and GCF. In my own words

Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite

### Warm Up answers. 1. x 2. 3x²y 8xy² 3. 7x² - 3x 4. 1 term 5. 2 terms 6. 3 terms

Warm Up answers 1. x 2. 3x²y 8xy² 3. 7x² - 3x 4. 1 term 5. 2 terms 6. 3 terms Warm Up Assignment 10/23/14 Section 6.1 Page 315: 2 12 (E) 40 58 (E) 66 Section 6.2 Page 323: 2 12 (E) 16 36 (E) 42 46 (E)

### Course Name: MAT 135 Spring 2017 Master Course Code: N/A. ALEKS Course: Intermediate Algebra Instructor: Master Templates

Course Name: MAT 135 Spring 2017 Master Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 01/15/2017 End: 05/31/2017 Course Content: 279 Topics (207

### Unit 3 Vocabulary. An algebraic expression that can contains. variables, numbers and operators (like +, An equation is a math sentence stating

Hart Interactive Math Algebra 1 MODULE 2 An algebraic expression that can contains 1 Algebraic Expression variables, numbers and operators (like +,, x and ). 1 Equation An equation is a math sentence stating

### Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper

Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,

### Study Guide for Math 095

Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

### Prep for College Algebra

Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)

### Chapter 3: Factors, Roots, and Powers

Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly

### Algebra I. Polynomials.

1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

### Controlling the Population

Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1

### Prep for College Algebra with Trigonometry

Prep for College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (246 topics +

### Math 0320 Final Exam Review

Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

### Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra

Course Title: College Preparatory Mathematics I Prerequisite: Placement with a score below 20 on ACT, below 450 on SAT, or assessing into Basic Applied Mathematics or Basic Algebra using Accuplacer, ASSET

### Answers of the MATH97 Practice Test Form A

Answers of the MATH97 Practice Test Form A A1) Answer B Section 1.2: concepts of solution of the equations. Pick the pair which satisfies the equation 4x+y=10. x= 1 and y=6 A2) Answer A Section 1.3: select

### LESSON 7.2 FACTORING POLYNOMIALS II

LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +

### Get Ready. 6. Expand using the distributive property. a) 6m(2m 4) b) 8xy(2x y) c) 6a 2 ( 3a + 4ab) d) 2a(b 2 6ab + 7)

Get Ready BLM 5 1... Classify Polynomials 1. Classify each polynomial by the number of terms. 2y x 2 + 3x + 2 c) 6x 2 y + 2xy + 4 d) x 2 + y 2 e) 3x 2 + 2x + y 4 6. Expand using the distributive property.

### Assignment #1 MAT121 Summer 2015 NAME:

Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also

### 7.2 Solving Quadratic Equations by Factoring

7.2 Solving Quadratic Equations by Factoring 1 Factoring Review There are four main types of factoring: 1) Removing the Greatest Common Factor 2) Difference of square a 2 b 2 3) Trinomials in the form

### LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6. POLYNOMIAL OPERATIONS I LESSON 6. POLYNOMIALS OPERATIONS I 63 OVERVIEW Here's what you'll learn in this lesson: Adding and Subtracting a. Definition of polynomial, term, and coefficient b. Evaluating

### The number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.

Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): To be able to add and subtract polynomials. Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number

### Section 9.1: Add and Subtract Polynomials. The number part of a term with a variable part.

Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number part of a term with a variable part. The

### Algebra One As of: September 2014 Teacher Contact: Ms.Zinn (CVHS-NGC)

Algebra One As of: September 2014 Teacher Contact: Ms.Zinn (CVHS-NGC) CCSS Unit Theme SKILLS ASSESSMENT & PRODUCTS Translate sentences into equations such as, The length of a rectangle is ten less than

### August 15, M1 1.4 Common Factors_Multiples Compacted.notebook. Warm Up MI 36. Jun 20 10:53 AM

Warm Up MI 36 8 14 18 Jun 20 10:53 AM 1 Assignment Jun 20 12:36 PM 2 Practice 7 13 A = bh 7 x 13 91 7 7 A = ½bh ½(7 x 7) ½(49) 24.5 Jun 20 12:36 PM 3 Practice 6 4 8 A=½bh 4 6x8 24 A=bh 4x8 32 4 5 8 8 A=bh

### Sections 7.2, 7.3, 4.1

Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals

### Algebra II Vocabulary Word Wall Cards

Algebra II Vocabulary Word Wall Cards Mathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development. The cards should