6.4 Factoring Polynomials


 Suzan Lindsey
 5 years ago
 Views:
Transcription
1 Name Class Date 6.4 Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Resource Locker Explore Analyzing a Visual Model for Polynomial Factorization Factoring a polynomial of degree n involves finding factors of lesser a degree that can be multiplied together to produce the polynomial. When a polynomial has degree 3, for example, you can think of it as a rectangular prism whose dimensions you need to determine. A The volumes of the parts of the rectangular prism are as follows: Red: V = x 3 Green: V = 2 x 2 Yellow: V = 8x Blue: V = 4 x 2 Total volume: V = x x 2 + 8x B The volume of the red piece is found by cubing the length of one side. What is the height of this piece? C D The volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height of the prism. Notice that the green prism shares two sides with the cube. What are the lengths of these two sides? What is the length of the third side of the green prism? E You showed that the width of the cube is and the width of the green prism is. What is the width of the entire prism? Module Lesson 4
2 You determined that the length of the green piece is x. Use the volume of the yellow piece and the information you have derived to find the length of the prism. Since the dimensions of the overall prism are x, x + 2, and x + 4, the volume of the overall prism can be rewritten in factored form as V = (x) (x + 2) (x + 4). Multiply these polynomials together to verify that this is equal to the original given expression for the volume of the overall figure. Reflect 1. Discussion What is one way to double the volume of the prism? Explain 1 Factoring Out the Greatest Common Monomial First Most polynomials cannot be factored over the integers, which means to find factors that use only integer coefficients. But when a polynomial can be factored, each factor has a degree less than the polynomial s degree. While the goal is to write the polynomial as a product of linear factors, this is not always possible. When a factor of degree 2 or greater cannot be factored further, it is called an irreducible factor. Example 1 A 6 x x 2 + 6x Factor each polynomial over the integers. 6 x x 2 + 6x Write out the polynomial. x (6 x x + 6) Factor out a common monomial, an x. 3x (2 x 2 + 5x + 2) Factor out a common monomial, a 3. x (2x + 1) (x + 2) Factor into simplest terms. Note: The second and third steps can be combined into one step by factoring out the greatest common monomial. Module Lesson 4
3 B 2 x 320x 3  x Write out the polynomial. ( x 210) Factor out the greatest common monomial. Reflect 2. Why wasn t the factor x 210 further factored? 3. Consider what happens when you factor x 210 over the real numbers and not merely the integers. Find a such that x 210 = (x  a) (x + a). Your Turn 4. 3 x x 2 + 4x Explain 2 Recognizing Special Factoring Patterns Remember the factoring patterns you already know: Difference of two squares: a 2  b 2 = (a + b) (a  b) Perfect square trinomials: a 2 + 2ab + b 2 2 = (a + b) and a 22ab + b 2 2 = (a  b) There are two other factoring patterns that will prove useful: Sum of two cubes : a 3 + b 3 2 = (a + b) (a  ab + b 2 ) Difference of two cubes : a 3  b 3 2 = (a  b) (a + ab + b 2 ) Notice that in each of the new factoring patterns, the quadratic factor is irreducible over the integers. Example 2 Factor the polynomial using a factoring pattern. A 27 x x Write out the polynomial. 27 x 3 3 = (3x) 3 64 = (4) Check if 27 x 3 is a perfect cube. Check if 64 is a perfect cube. a 3 + b 3 = (a + b) (a 2  ab + b 2 ) Use the sum of two cubes formula to factor. (3x) = (3x +4) ( (3x) 2  (3x) (4) ) 27 x = (3x + 4) (9 x 212x + 16) Module Lesson 4
4 B 8 x Write out the polynomial. 8 x 3 = ( x) 3 Check if 8 x 3 is a perfect cube. 27 = ( ) 3 Check if 27 is a perfect cube. a 3  b 3 = (a  b) ( a 2 + ab + b 2 ) Use the difference of two cubes formula to factor. 8 x 327 = ( x  ) ( x 2 + x + ) Reflect 5. The equation 8 x 327 = 0 has three roots. How many of them are real, what are they, and how many are nonreal? Your Turn x 4 + 5x Module Lesson 4
5 Explain 3 Factoring by Grouping Another technique for factoring a polynomial is grouping. If the polynomial has pairs of terms with common factors, factor by grouping terms with common factors and then factoring out the common factor from each group. Then look for a common factor of the groups in order to complete the factorization of the polynomial. Example 3 Factor the polynomial by grouping. A x 3 + x 2 + x + 1 Write out the polynomial. x 3  x 2 + x  1 Group by common factor. ( x 3  x 2 ) + (x  1) Factor. x 2 (x  1) + 1 (x  1) Regroup. ( x 2 + 1) (x  1) B x 4 + x 3 + x + 1 Write out the polynomial. x 4 + x 3 + x + 1 Group by common factor. ( + ) + (x + 1) Factor. (x + 1) + (x + 1) Regroup. ( + )(x + 1) Apply sum of two cubes to the first term. (  + 1) (x + 1) (x + 1) Substitute this into the expression and simplify. ( ) 2 ( ) Your Turn 7. x x 2 + 3x + 2 Explain 4 Solving a RealWorld Problem by Factoring a Polynomial Remember that the zeroproduct property is used in solving factorable quadratic equations. It can also be used in solving factorable polynomial equations. Module Lesson 4
6 Example 4 Write and solve a polynomial equation for the situation described. A A water park is designing a new pool in the shape of a rectangular prism. The sides and bottom of the pool are made of material 5 feet thick. The length must be twice the height (depth), and the interior width must be three times the interior height. The volume of the box must be 6000 cubic feet. What are the exterior dimensions of the pool? The dimensions of the interior of the pool, as described by the problem, are the following: h = x  5 w = 3x  15 l = 2x  10 The formula for volume of a rectangular prism is V = lwh. Plug the values into the volume equation. V = (x  5) (3x  15) (2x  10) V = (x  5) (6 x 260x + 150) V = 6 x 390 x x Now solve for V = = 6 x 390 x x = 6 x 390 x x Factor the resulting new polynomial. 6 x 390 x x = 6 x 2 (x  15) (x  15) = (6 x ) (x  15) The only real root is x = 15. B The interior height of the pool will be 10 feet, the interior width 30 feet, and the interior length 20 feet. Therefore, the exterior height is 15 feet, the exterior length is 30 feet, and the exterior width is 40 feet. Engineering To build a hefty wooden feeding trough for a zoo, its sides and bottom should be 2 feet thick, and its outer length should be twice its outer width and height. What should the outer dimensions of the trough be if it is to hold 288 cubic feet of water? Volume = Interior Length(feet) Interior Width(feet) Interior Height(feet) 288 = (  4) (  4) (  2) 288 = x 3  x 2 + x  0 = x 3  x 2 + x  0 = (x  ) + (x  ) Image Credits: morrison77/shutterstock 0 = ( x 2 + ) (x  ) The only real solution is x =. The trough is feet long, feet wide, and feet high. Module Lesson 4
7 Your Turn 8. Engineering A new shed is being designed in the shape of a rectangular prism. The shed s side and bottom should be 3 feet thick. Its outer length should be twice its outer width and height. What should the outer dimensions of the shed be if it is to have 972 cubic feet of space? Elaborate 9. Describe how the method of grouping incorporates the method of factoring out the greatest common monomial. 10. How do you decide if an equation fits in the sum of two cubes pattern? 11. How can factoring be used to solve a polynomial equation of the form p (x) = a, where a is a nonzero constant? 12. Essential Question CheckIn What are two ways to factor a polynomial? Module Lesson 4
8 Evaluate: Homework and Practice Factor the polynomial, or identify it as irreducible. 1. x 3 + x 212x 2. x Online Homework Hints and Help Extra Practice 3. x x x 2 + 6x 5. 8 x x 3 + 6x x x x x x x x x x 12. x Module Lesson 4
9 Factor the polynomial by grouping. 13. x x 2 + 6x x x 2  x x x x x x 364x x x 2 + 3x x 44 x 3  x + 1 Write and solve a polynomial equation for the situation described. 19. Engineering A new rectangular outbuilding for a farm is being designed. The outbuilding s side and bottom should be 4 feet thick. Its outer length should be twice its outer width and height. What should the outer dimensions of the outbuilding be if it is to hold 2304 cubic feet? Image Credits: Alex Ramsay/Alamy 20. Arts A piece of rectangular crafting supply is being cut for a new sculpture. You want its length to be 4 times its height and its width to be 2 times its height. If you want the wood to be 64 cubic centimeters, what will its length, width, and height be? Module Lesson 4
10 21. Engineering A new rectangular holding tank is being built. The tank s side and bottom should be 1 foot thick. Its outer length should be twice its outer width and height. What should the outer dimensions of the tank be if it is to hold 36 cubic feet? 22. Construction A piece of granite is being cut for a building foundation. You want its length to be 8 times its height and its width to be 3 times its height. If you want the granite to be 648 cubic yards, what will its length, width, and height be? 23. State which, if any, special factoring pattern each of the following polynomial functions follows: a. x 24 b. 3 x c. 4 x d. 16 x e. 64 x 3  x H.O.T. Focus on Higher Order Thinking 24. Communicate Mathematical Ideas What is the relationship between the degree of a polynomial and the degree of its factors? 25. Critical Thinking Why is there no sumoftwosquares factoring pattern? Image Credits: Gennadiy Iotkovskiy/Alamy Module Lesson 4
11 26. Explain the Error Jim was trying to factor a polynomial function and produced the following result: 3 x 3 + x 2 + 3x + 1 Write out the polynomial. 3 x 2 (x + 1) + 3 (x + 1) Group by common factor. 3 ( x 2 + 1) (x + 1) Regroup. Explain Jim s error. 27. Factoring can also be done over the complex numbers. This allows you to find all the roots of an equation, not just the real ones. Complete the steps showing how to use a special factor identity to factor x over the complex numbers. x Write out the polynomial. x 2  (4) (x + ) (x  ) Factor. (x + 2i) ( ) Simplify. 28. Find all the complex roots of the equation x 416 = Factor x 3 + x 2 + x + 1 over the complex numbers. Module Lesson 4
12 Lesson Performance Task Sabrina is building a rectangular raised flower bed. The boards on the two shorter sides are 6 inches thick, and the boards on the two longer sides are 4 inches thick. Sabrina wants the outer length of her bed to be 4 times its height and the outer width to be 2 times its height. She also wants the boards to rise 4 inches above the level of the soil in the bed. What should the outer dimensions of the bed be if she wants it to hold 3136 cubic inches of soil? Image Credits: Gary K Smith/Alamy Module Lesson 4
Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.
5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological
More informationFactor Polynomials Completely
9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationVeterans Upward Bound Algebra I Concepts  Honors
Veterans Upward Bound Algebra I Concepts  Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More informationLesson 11: Volume with Fractional Edge Lengths and Unit Cubes
Lesson : Volume with Fractional Edge Lengths and Unit Cubes Student Outcomes Students extend their understanding of the volume of a right rectangular prism with integer side lengths to right rectangular
More informationVOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.
Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More information10 7, 8. 2. 6x + 30x + 36 SOLUTION: 89 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.
Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More information6.3 FACTORING ax 2 bx c WITH a 1
290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationFactoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns
Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial  2x 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationSOL WarmUp Graphing Calculator Active
A.2a (a) Using laws of exponents to simplify monomial expressions and ratios of monomial expressions 1. Which expression is equivalent to (5x 2 )(4x 5 )? A 9x 7 B 9x 10 C 20x 7 D 20x 10 2. Which expression
More informationArea of Parallelograms (pages 546 549)
A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular
More information1.3 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.
More informationHow To Solve Factoring Problems
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationSect 6.7  Solving Equations Using the Zero Product Rule
Sect 6.7  Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationLesson 13: The Formulas for Volume
Student Outcomes Students develop, understand, and apply formulas for finding the volume of right rectangular prisms and cubes. Lesson Notes This lesson is a continuation of Lessons 11, 12, and Module
More informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationPolynomial Equations and Factoring
7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationFactoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
More informationUnit 3: Day 2: Factoring Polynomial Expressions
Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More information15.1 Factoring Polynomials
LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE
More informationLearning Objectives 9.2. Media Run Times 9.3
Unit 9 Table of Contents Unit 9: Factoring Video Overview Learning Objectives 9.2 Media Run Times 9.3 Instructor Notes 9.4 The Mathematics of Factoring Polynomials Teaching Tips: Conceptual Challenges
More informationFACTORING OUT COMMON FACTORS
278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the
More informationMathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework
Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010  A.1 The student will represent verbal
More informationGeorgia Department of Education Georgia Standards of Excellence Framework GSE Grade 6 Mathematics Unit 5
**Volume and Cubes Back to Task Table In this problembased task, students will examine the mathematical relationship between the volume of a rectangular prism in cubic units and the number of unit cubes
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationPolynomials and Factoring; More on Probability
Polynomials and Factoring; More on Probability Melissa Kramer, (MelissaK) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
More informationArea is a measure of how much space is occupied by a figure. 1cm 1cm
Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number
More informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationFactoring a Difference of Two Squares. Factoring a Difference of Two Squares
284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More informationKeystone Exams: Algebra I Assessment Anchors and Eligible Content. Pennsylvania
Keystone Exams: Algebra I Assessment Anchors and Pennsylvania Algebra 1 2010 STANDARDS MODULE 1 Operations and Linear Equations & Inequalities ASSESSMENT ANCHOR A1.1.1 Operations with Real Numbers and
More informationGeometry Notes VOLUME AND SURFACE AREA
Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate
More informationG r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e  C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam
G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e  C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d
More informationGAP CLOSING. Volume and Surface Area. Intermediate / Senior Student Book
GAP CLOSING Volume and Surface Area Intermediate / Senior Student Book Volume and Surface Area Diagnostic...3 Volumes of Prisms...6 Volumes of Cylinders...13 Surface Areas of Prisms and Cylinders...18
More informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationArea of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in
Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in
More informationWhat You ll Learn. Why It s Important
These students are setting up a tent. How do the students know how to set up the tent? How is the shape of the tent created? How could students find the amount of material needed to make the tent? Why
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationThe majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationMeasurement. Volume It All Stacks Up. Activity:
Measurement Activity: TEKS: Overview: Materials: Grouping: Time: Volume It All Stacks Up (7.9) Measurement. The student solves application problems involving estimation and measurement. The student is
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationLesson 1: Multiplying and Factoring Polynomial Expressions
Lesson 1: Multiplying and Factoring Polynomial Expressions Student Outcomes Students use the distributive property to multiply a monomial by a polynomial and understand that factoring reverses the multiplication
More informationSolving Geometric Applications
1.8 Solving Geometric Applications 1.8 OBJECTIVES 1. Find a perimeter 2. Solve applications that involve perimeter 3. Find the area of a rectangular figure 4. Apply area formulas 5. Apply volume formulas
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More informationHow To Factor By Gcf In Algebra 1.5
72 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
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationMATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
More informationTopic: Special Products and Factors Subtopic: Rules on finding factors of polynomials
Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:
More informationDeterminants can be used to solve a linear system of equations using Cramer s Rule.
2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More information2x 2x 2 8x. Now, let s work backwards to FACTOR. We begin by placing the terms of the polynomial inside the cells of the box. 2x 2
Activity 23 Math 40 Factoring using the BOX Team Name (optional): Your Name: Partner(s): 1. (2.) Task 1: Factoring out the greatest common factor Mini Lecture: Factoring polynomials is our focus now. Factoring
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More information3.3. The Factor Theorem. Investigate Determining the Factors of a Polynomial. Reflect and Respond
3.3 The Factor Theorem Focus on... factoring polynomials explaining the relationship between the linear factors of a polynomial expression and the zeros of the corresponding function modelling and solving
More information6706_PM10SB_C4_CO_pp192193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More information