Algebra 1 Unit 6B Factoring

Similar documents
Chapter 5: Exponents and Polynomials

Algebra I. Exponents and Polynomials. Name

Algebra I. Polynomials.

Algebra I Polynomials

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

Multiplication of Polynomials

Collecting Like Terms

LESSON 9.1 ROOTS AND RADICALS

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

Math 10-C Polynomials Concept Sheets

Assignment #1 MAT121 Summer 2015 NAME:

UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name:

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

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

Name: Teacher: Per: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10. Unit 8. [Polynomials] Unit 8 Polynomials 1

Unit 3 Factors & Products

MATHEMATICS 9 CHAPTER 7 MILLER HIGH SCHOOL MATHEMATICS DEPARTMENT NAME: DATE: BLOCK: TEACHER: Miller High School Mathematics Page 1

MATH98 Intermediate Algebra Practice Test Form A

Mathwithsheppard.weebly.com

Algebra 2. Factoring Polynomials

Maintaining Mathematical Proficiency

Solving Multi-Step Equations

Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to

POLYNOMIAL: A polynomial is a or the

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

Algebra I Unit Report Summary

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals

Polynomials 370 UNIT 10 WORKING WITH POLYNOMIALS. The railcars are linked together.

Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

27 Wyner Math 2 Spring 2019

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

Algebra I. Slide 1 / 216. Slide 2 / 216. Slide 3 / 216. Polynomials

mn 3 17x 2 81y 4 z Algebra I Definitions of Monomials, Polynomials and Degrees 32,457 Slide 1 / 216 Slide 2 / 216 Slide 3 / 216 Slide 4 / 216

Algebra 1 Unit 6 Notes

5.3. Polynomials and Polynomial Functions

Classifying Polynomials. Simplifying Polynomials

When factoring, we ALWAYS start with the (unless it s 1).

ACTIVITY: Factoring Special Products. Work with a partner. Six different algebra tiles are shown below.

When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

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

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

CHAPTER 1 POLYNOMIALS

Rising 8th Grade Math. Algebra 1 Summer Review Packet

Polynomials. This booklet belongs to: Period

Unit 13: Polynomials and Exponents

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

Lesson 3: Polynomials and Exponents, Part 1

LESSON 7.2 FACTORING POLYNOMIALS II

PRE-ALGEBRA SUMMARY WHOLE NUMBERS

LESSON 6.2 POLYNOMIAL OPERATIONS I

Math 46 Final Exam Review Packet

Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring

Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook

Example #3: 14 (5 + 2) 6 = = then add = 1 x (-3) then. = 1.5 = add

9-1 Skills Practice Factors and Greatest Common Factors Find the factors of each number. Then classify each number as prime or composite

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

Quadratic Expressions and Equations

My Math Plan Assessment #1 Study Guide

Adding and Subtracting Polynomials

8-1 Factors and Greatest Common Factors 8-1. Factors and Greatest Common Factors

7.7. Factoring Special Products. Essential Question How can you recognize and factor special products?

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Beginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College

Remember, you may not use a calculator when you take the assessment test.

Prime Factorization and GCF. In my own words

Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6

Chapter 8 Polynomials and Factoring

Simplify each numerical expression. Show all work! Only use a calculator to check. 1) x ) 25 ( x 2 3) 3) 4)

SECTION 1.4 PolyNomiAls feet. Figure 1. A = s 2 = (2x) 2 = 4x 2 A = 2 (2x) 3 _ 2 = 1 _ = 3 _. A = lw = x 1. = x

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College

Properties of Real Numbers

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

Grade 9 Mathematics Unit #2 Patterns & Relations Sub-Unit #1 Polynomials

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

LESSON 6.2 POLYNOMIAL OPERATIONS I

1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.

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

We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:

Review Notes - Solving Quadratic Equations

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

Lesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality

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

How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials

5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014. c = Properites of Exponents. *Simplify each of the following:

2009 Math Olympics Level II

8-1: Adding and Subtracting Polynomials

Section 6.5 A General Factoring Strategy

LESSON 6.3 POLYNOMIAL OPERATIONS II

Quick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors.

Unit 5 Quadratic Expressions and Equations

Prerequisites. Copyright Cengage Learning. All rights reserved.

Math 0320 Final Exam Review

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

3 According to the standard convention for exponentiation,

Combining Like Terms in Polynomials

Math 75 Mini-Mod Due Dates Spring 2016

Transcription:

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 Feb. 16 17 B Day 18 A Day 19 B Day 20 A Day No School Staff Development Factoring Trinomials Factoring with Patterns GCF difference of squares perfect square trinomials Dividing Polynomials Retest CBA #4 Quiz Factoring 23 B Day 24 A Day 25 B Day 26 A Day 27 B Day Divide Polynomials Quiz Factoring Elaboration day Test CBA #6 (grade will need to go on the NEXT six weeks marking period) 1

WARM-UP # Simplify each expression. 1. (x + 4)(x 6) 2. (10x 2 + 5x 6) (8x 2 2x + 7) 3. 5x 2 (2xy 3x) 4. (2x + 5)(3x + 6) 5. (x 2 + y 2 ) (-x 2 + y 2 ) 6. a b ab 4 2 2 6 2

Notes GCF and Factoring Prime Number a whole number, greater than 1, whose only factors are 1 and itself Composite Number a whole number, greater than 1, that is not prime Prime Factorization a whole number expressed as a product of factors are all prime numbers (i.e. factor tree) Greatest Common Factor (GCF) the greatest common factor of two or more integers is the greatest number that is a factor of all the integers EX1: State whether each number is prime or composite. If the number if composite, find its prime factorization (tree). a. 28 b. 61 c. 112 d. 150 EX2: Find the GCF between two numbers using the calculator. a. -45, 15 b. 169, 13 c. -20, 440 d. 96, 12, -8 Greatest common factor for the same variable will be LOWEST exponent of that given variable. Factoring to express a polynomial as the product of a monomial and a polynomial EX3: Find the GCF for each set of monomials. a. x 2, x 5, x 4 b. 49x, 343x 2 c. 4a 7 b, 28ab d. 96y, 12x, -8y 3

EX4: Factor each polynomial. Notes GCF and Factoring a. 24w + 72z b. 30ab 2 + a 2 b 12ac 3 c. x 4 18x 2 + 22x d. a + 10a 2 b 3 e. 88x 4 11x 7 + 66x 5 f. 14c 3 42c 5 49c 4 g. 48w 2 x + 18wx 2 36wx h. -x 5 4x 4 + 23x 3 x 6 i. 8x 7y + w j. 18y 2 50 k. x 3 + 2x 2 + x 4

Reverse Distribution Find a monomial and a trinomial whose product is equal to each problem below. Cut and paste it in the correct place. Problems Monomials (GCF) Trinomials 1. 12x 2 + 3x 6 2. 12x 4 6x 2 + 3x 3. 24x 5 + 12x 4 4x 3 4. 4x 4 12x 3 + 6x 2 5. 6x 3 24x 2 12x 6. 10a 4 b 2 5a 3 b + a 2 b 7. 5a 6 b 5 + 10a 5 b 4 15a 4 b 3 8. 20a 5 b 5 + 10a 4 b 4 30a 3 b 3 9. 50a 6 b 2 30a 5 b 3 + 10a 4 b 4 10. 50a 7 b 6 15a 5 b 2 + 25a 3 5

Monomials (GCF) Trinomials 6x (10a 2 b 5a + 1) 5a 4 b 3 (10a 4 b 6 3a 2 b 2 + 5) 10a 4 b 2 (a 2 b 2 + 2ab 3) 5a 3 (4x 2 + x 2) 3 (x 2 4x 2) 3x (2x 2 6x + 3) a 2 b (2a 2 b 2 + ab 3) 10a 3 b 3 (6x 2 + 3x 1) 4x 3 (5a 2 3ab + b 2 ) 2x 2 (4x 3 2x + 1) 6

Name Date GCF and Factoring Factor out the GCF. 1. x 3 + x 2 + x 2. 15a + 12b + 6c 3. 8x 2 18y 2 4. x 2 y 2y 5. z 3 + 4z 6. 4x 2 4x 7. 15x 2 50x 10 8. 12a 11b 9. 64c 3 56c 2 + 88c 10. 24x 6 y 3 32x 3 y 2 20x 2 y 2 11. 2x 4 + 24x 2 12. 12x 3 y 4 40xy 5 7

Simplify each expression. 13. (2x + 5xy + 7y) + (3x + 7xy + y) 14. (3x + 2y) (5x + 6y) 40a b 15. (a + b) 0 16. 5 9 20a b 1 7 17. (2m -4 n 3 )(-5mn -7 ) 18. a b 1 2 19. Find the volume of a cylinder with a diameter of 4x 3 y and a height of 7x 2 y 4. 20. Find the volume of a cube with sides 2b 3 r 2. Solve. 21. 4x + 2 = 2(5x 11) 22. 9 4x < 10 8

WARM-UP # Find the missing information on the given rectangles. 17 12 What is the area? This is the same size rectangle just divided up. 5 12 What is the area of the first rectangle? 12 What is the area of the second rectangle? What is the area of the whole rectangle? Write the area of each rectangle inside each box for both of the rectangle below and answer the questions. 5 12 3 What is the total area of the rectangle? 9 6 2 15 What is the total area of the rectangle? 6 What do you notice about all of the rectangles above? What is special about the length and width? What is the length and width of all the rectangles? 9

EXPLORE Given the rectangles below, determine the length and width of each rectangle and the area. Rectangles are not drawn to scale. 2 5 5 Total Area 14 12 Total Area 6 Length Width 21 Length Width 20 12 Total Area Length 2x 5x Total Area Length 21 Width 10 25 Width 3x 2 2x Total Area Length 4x 2x 1 Total Area Length 30x 20 Width -1 Width x 2 Total Area Length x x 2-4 Total Area Length 30x 20 Width -12 Width Total Area Length (x 1) Width (2x + 3) 10

Notes Factoring Trinomials EX1. Recall the box method to multiply two binomials. Multiply (x 3)(x + 2). Factors: Product: EX2. Find the missing dimension of each trinomial s box. Fill in the blank cells in each box. a. a 2 + 7a + 10 = (a + 5)( ) b. c 2 10c + 21 = (c 3)( ) a a 2 c 2 c -3 5 10 21 c. y 2 2y 15 = (y + 3) ( ) d. n 2 + 3n 28 = (n 4) ( ) y y 2 n 2 n -4 3-15 -28 e. How do the quantities you filled in the 2 blank cells relate to the original trinomial? 11

Notes Factoring Trinomials EX3. Write the numbers that give a sum of 5x and a product of 50x 2. Standard form: EX4. Factor each trinomial. a. x 2 + 7x + 10 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) b. x 2 + 3x 4 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) c. x 2 64 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) 12

Notes Factoring Trinomials d. 3x 2 + 14x + 8 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) e. 2y 2 7y + 6 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) f. 6x 2 21x 12 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) 13

Notes Factoring Trinomials Factoring Using Algebra Tiles EX5. Determine the factors of each polynomial. a. b. c. d. 14

Name Date 11 15 10 11 1 8 16 7 17 12 5 11 2 16 17 6 4 3 14 8 7 9 6 4 3 14 16 13 A B C D E (x 2)(5x 8) (2x + 3)(3x 2) (x 13)(x + 3) (x + 2)(x 3) (x 2)(x + 3) F G H I J (x 8)(2x + 5) (x + 2)(7x + 3) (x 2)(x + 1) (x 5)(x + 3) (x + 5)(x 3) K L M N O (x 7)(x + 7) (x 2)(x 8) (x + 2)(x + 8) (x 7)(x 2) (x + 7) 2 P Q R S T (x 7)(x + 2) (x + 2)(x 8) (x 3)(x + 3) (x 6 )(2x 1) (x + 6) 2 U V W X Y (x 5)(x + 5) (x 5)(x 5) (x 3) 2 (x 3)(x + 20) (x 6)(x 2) Z (x + 9)(x + 7) Factoring Trinomials Directions: Match that answer to the correct letter of the alphabet. Enter that letter of the alphabet on the blank corresponding to the problem number. Factor each polynomial make sure to show your work. 1. x 2 49 2. x 2 + 12x + 36 3. 7x 2 + 17x + 6 4. x 2 9x + 14 5. 5x 2 18x + 16 6. x 2 2x 15 15

7. x 2 25 8. x 2 8x + 12 9. 2x 2 13x + 6 10. x 2 + x 6 11. x 2 10x 39 12. 2x 2 11x 40 13. x 2 + 17x 60 14. 6x 2 + 5x 6 15. x 2 x 2 16. x 2 + 14x + 49 17. x 2 9 Identify the simplified area of each rectangle. Then determine the factors. 18. 19. 20. 16

WARM-UP # 1. Factor: x 2 + 13x + 12 Factors: What did you notice? 2. Factor: 4x 2 9 Factors: What did you notice? 3. Factor: x 2 + 6x + 9 Factors: What did you notice? 17

18

Explain Factoring with Patterns Difference of Squares a 2 b 2 = (a) 2 (b) 2 = (a + b)(a b) difference opposite signs Conjugate pairs *Warning: a 2 + b 2 does not factor To recognize perfect squares, look for coefficients that are squares of integers and variables raised to even powers. EX1: Factor, if possible, using the difference of squares. a. 4x 2 9y 2 b. a 2 16b 2 c. 9x 4 25y 4 d. u 2 v 2 w 2 z 2 e. 25m 2 + 36n 2 19

Explain Factoring with Patterns Perfect Square Trinomials a 2 + 2ab + b 2 = (a + b)(a + b) = (a + b) 2 a 2 2ab + b 2 = (a b)(a b) = (a b) 2 EX2: Factor each of the following. a. x 2 + 6x + 9 b. x 2 10x + 25 c. a 2 + 8a + 16 d. 9a 2 24a + 16 20

Name Date Factoring Patterns Determine whether each statement is TRUE. If not, find the correct product. 1. (3x + 1) 2 = 9x 2 + 6x + 1 2. (m 4) 2 = m 2 16m + 16 3. (5t 2) 2 = 25t 2 20t + 4 4. (2n + 7) 2 = 4n 2 + 28n + 49 5. (2b + 3) 2 = 4b 2 + 12b + 6 6. (2a + b) 2 = 4a 2 + 4ab + b 2 Factor each polynomial. If it cannot be factored, write prime. 7. t 2 12t + 36 8. a 2 + 2ab + b 2 9. 4t 2 + 20t + 25 10. n 2 1 11. 144 25n 2 12. 16a 2 24ab + 9b 2 21

13. t 2 18t + 81 14. 4n 2 9 15. 25 + 10t + t 2 16. n 2 49 17. 49m 2 16n 2 18. a 2 8a + 64 19. 49a 2 + 14a + 1 20. 81 121n 2 21. Which is the correct factorization of 45x 2 + 20y 2? A. 5(3x + 2y) 2 B. 5(3x 2y) 2 C. 5(3x + 2y)(3x 2y) D. 5(3x + 2y)(3x 2y) 22. Challenge Determine the value(s) of k for which each expression is a perfect square trinomial. a. 49x 2 84k + k b. 4x 2 + kx + 9 22

Explore Dividing Polynomials Remember when we MULTIPLIED: (using a box) (x + 2)(x + 6) or (2y + 1)(3y 4) So can you now DIVIDE these polynomials: (using a box) x 2 + 8x + 12 x + 2 6y 2 5y 4 3y 4 3y 4 x +2 So.. x 2 + 8x + 12 x + 2 the quotient is: 6y 2 5y 4 3y 4 the quotient is: 23

24

Explain Dividing Polynomials Dividing is the opposite operation of. Therefore, we will use the to assist in dividing trinomials when given a trinomial divided by a binomial. EX1. Simplify each expression. a. x 2 + 4x 5 x 1 Quotient: x x 2 5x 1 x 5 b. 2x 2 + 11x + 12 x + 4 Quotient: 2x 2 8x 3x 12 c. 6x 2 + 13x 5 3x 1 Quotient: 25

Explain Dividing Polynomials EX2. Simplify each expression ON OUR OWN. a. 5x 2 + 33x 14 x + 7 Quotient: b. x 2 11x + 24 x 8 Quotient: c. 2x 2 + 17x 9 2x 1 Quotient: 26

Name Gingerbread Man Date Divide each polynomial. Each answer determines the next location of the traveling gingerbread man. Determine the path the gingerbread man makes through the school. 1. x 2 + 12x + 27 x + 9 2. x 2 13x + 40 x 8 3. x 2 7x 44 x + 4 4. x 2 + x 42 x 6 5. 2x 2 + 9x + 4 2x + 1 6. 2x 2 15x + 7 x 7 7. 2x 2 + 7x 15 2x 3 8. 6x 2 + 17x + 5 2x + 5 9. 3x 2 + 19x 14 x + 7 10. 12x 2 + 7x 12 3x + 4 11. 8x 2 + 2x 3 2x 1 12. 9x 2 15x + 4 3x 4 27

Front Door Cafeteria (3x 1) Principal Counselor Nurse (x 2) (4x + 3) (x + 11) Secretary (4x 3) Attendance (x + 1) Trophy case (3x 2) Teacher Workroom K - 2 Kindergarte n 1 st Grade 2 nd Grade (x + 2) (2x 1) (3x + 1) Library 3 rd Grade (x 4) (x + 5) Playground 2 (x + 6) (x 5) 4 th Grade 5 th Grade (3x + 2) (x + 4) Teacher Workroom 3 5 Theater (x + 3) Home Ec Lab Where s my class? Computer Lab (x 11) Art Room (x + 7) Playground 1 (x 7) 28

Name Date 1. Find the volume of a cube with sides 2b 3 r 2. Review CBA #6 2. Find the volume of a cylinder that has a radius of 5s 3 t 5 and a height of 2s 2 t 4. 3. Find the area of a triangle that has a base of 32mn 7 and a height of 3m 4 n 3. 4. If a rectangle has an area of 16x 7 y 4 and a length of 4x 3 y, what is its width? 5. Distance (d), rate (r), and time (t) are related by the formula d = rt. If a ball rolls 36p 4 q 9 feet for 4p 2 q 3 minutes, what is the rate? 6. Write an expression that best represents the area of a square with sides of 7x 4 y 3? 7. Find the perimeter and area of the rectangle in terms of n. 3n 5 2n + 10 29

8. Find the perimeter and area of the triangle in terms of x. 3n + 5 5n 1 2n Simplify each expression. 9. (-2x + x 2 ) x(5x 4) + (9x 2 6x) 10. p(2p 3) + (p 3)(4p + 1) 11. (3x 5 ) 3 (2x 7 ) 2 12. (-3x 6 ) 2 13. (3r + 7) 2 14. 8 7 12x y z 15. 2 6 5 4x y z ( 2a b ) ( 7ab ) 10 4 3a b 4 2 3 9 0 16. n 6 + n + n 6 17. The dimensions of a wall are 7xy feet by 8x 2 y 3 feet. A picture has dimensions 2x feet by x 2 y 4 feet. If the picture is hanging on the wall as shown, what is the area of the wall not covered by the picture? 18. A pitcher contains 16x 5 y 4 ounces of water. A mug holds 2x 2 y ounces. Leticia pours water from the full pitcher into mugs. If she filled ax b y c mugs, what is the value of a + b + c? 30

19. Find the area of a circle with radius 6r 3 s 5 inches. 20. Find the area of a rectangle with side lengths (x 2 7x) and (2x 2 + 3x + 1). 21. Describe and correct the error in finding the product of the given polynomials. 22. The area of a rectangle is 3x 2 10x 8. Find the dimensions (length and width) of the rectangle. Factor out the greatest common monomial factor. 23. 16a 2 40b 24. -36s 3 + 18s 2 54s 25. 17abc 2 6a 2 c 31

Completely factor each of the following polynomials. 26. r 2 + 2r 24 27. y 2 2y 15 28. 2x 2 + 12x + 16 29. 2ax 2 3ax 35a 30. 4a 2 + 9a 9 31. 6k 2 + 13k + 6 32. 9x 2 121 33. k 2 49 34. 64u 2 25 Identify the simplified area of each rectangle. Then determine the factors. 35. 36. 37. 32

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback