Chapter 11: Factoring Polynomials Greatest Common Factor. ca d Factoring When Terms Have a Common Factor. Factor completely. B Å'B * # "& B Ä'B
|
|
|
- Ethel Harris
- 6 years ago
- Views:
Transcription
1 Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes Spring Greatest Common Factor ca d Factoring When Terms Have a Common Factor 1. Factor completely. B Å'B * "& 2. B Ä B $ ( 3. B ÅB 4. B Ä'B % % $ 5. )B Å%B 6. 'B Å"!B Ä$B "% " "! ) 7. B Ä B 8. B Ä B Ä B Ä B & & $ $ $ $ % $ ' & % $
2 9. B a bä a b 10. B a bä a b 11. B abå& bä abå& b 12. B abä" bä abä" b cb d Factoring by Grouping: Four Terms Factor by grouping. $ $ 13. ": Å"': Ä*:Å" 14. )B Å"B Ä'BÅ* Factor by grouping: $ $ 15. $B ÄB Å"&BÅ"! 16. B Å$B Å(BÄ" Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 2 of 15
3 11.2 Factoring Trinomials of the Form B Ä,BÄ- cd c Factoring Trinomials: Signs Only 17. B ÅBÅ"& ú ab $ bab & b 18. B Å'BÄ*ú ab $ bab $ b 19. B Ä'BÄ&ú ab " bab & b 20. B Ä*BÄ"% ú ab bab ( b 21., Ä,Å)ú ab bab % b 22., Ä&,Ä%ú a, " ba, % b " " " " " " 23. B Ä BÄ ú åb çåb ç 24. B Ä BÄ ú åb çåb ç & & & & $ * $ $ 25.. Å(. Ä" ú ab $ bab % b 26.. Å(. Ä"! ú a. ba. & b 27. C Å(C Ä"! ú ab bab & b 28. C Å""C Ä"! ú ac " bac "! b 29. B Ä(BÅ") ú ab bab * b 30. B Å"%BÅ"& ú ab " bab "& b 31. B ÄBÅ"& ú ab $ bab & b 32. B Å'B Å"' ú ab bab ) b 33. C Ä$C Å"! ú ac bac & b 34. C Å%C Å%& ú ac * bac & b 35. B ÄBÅ** ú ab * bab "" b 36. B ÅBÅ** ú ab * bab "" b % Å+ Å$& ú à + ( âà + & â Ä+Å$& ú a+ ( ba+ & b Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 3 of 15
4 cd d Factoring Trinomials: B Ä,BÄ- Factor completely. 39. B Ä"BÄ$& 40. B ÅBÅ" 41. B Ä%BÅ%& 42. B Ä"!BÅ% 43. B Ä%BÅ'! 44. B Ä$BÄ 45. B Ä)BÄ"' 46. B Ä"!BÄ& 47. B Å"%BÄ%* 48. B Å'BÄ* 49. B Ä!BÅ%* 50. B Ä!BÅ* 51. B Ä!BC Å"'C 52. %B Ä!BC Å%*C Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 4 of 15
5 cd e Factoring Trinomials with GCF : 5B Ä5,BÄ5- Factor completely: Factor completely: 53. $ B Å'B Å"'B 54. $ C Å%C Å%&C $ 55. B Ä(B Ä"B 56. B Ä)BÄ*' $ 57. $B Ä"B Å"$&B 58. &B CÄ&!BC Å"!C Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 5 of 15
6 11.3/4 Factoring Trinomials of the Form +B Ä,BÄ- where + " Review: Factor each polynomial by grouping. 59. B Ä&BÄBÄ"! 60. B ÄBÄ(BÄ"% 61. )B Å'BÅ)BÄ" 62. $B Å%BÅ"BÄ"' % 63. B Å'B Å&B Ä"& 64. %B Ä'BÅ'BÅ* % % 65. *B Å'B Å'B Ä% 66. B Ä'B Ä&B Ä"& Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 6 of 15
7 cd f Factoring Trinomials by Grouping: +B Ä,BÄ- a ba b ú a ba b ú 67. &B ÅBÅ") 68. B Å(BÅ% a bäa b ú a bäa b ú a ba b ú a ba b ú 69. $B ÄBÅ% 70. $B Ä%BÅ"& a bäa b ú a bäa b ú a ba b ú a ba b ú 71. 'B Ä"$BÄ' 72. 'B Ä$BÄ( a bäa b ú a bäa b ú Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 7 of 15
8 a ba b ú a ba b ú 73. (B Ä"&BÄ 74. $B Ä%BÄ" a bäa b ú a bäa b ú a ba b ú a ba b ú 75. *B Ä'BÅ) 76. %B Ä%BÅ"& a bäa b ú a bäa b ú a ba b ú a ba b ú 77. "&B Ä"*BÅ"! 78. B ÄBÅ" a bäa b ú a bäa b ú Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 8 of 15
9 11.5 Factoring Perfect Squares and Difference of Squares cg d Recognizing Trinomial Squares Then (Multiplying)... Now (Factoring)... a+ä, b ú+ Ä+,Ä, + Ä+,Ä, ú a+ä, b a+å, b ú+ Å+,Ä, + Å+,Ä, ú a+å, b How to recognize a Trinomial Square. E Two terms, + and,, must be squares, such F G) % as %fl Bfl&B fl"'>. There must be no minus sign before + or,. If we multiply + and, and double the result, we get eitherthe remaining term +,, or its opposite, Å+,. ch d Factoring Trinomial Squares 79. Factor: B Ä'BÄ* 80. Factor: B Å"%BÄ%* 81. Factor: B Ä%*Å"%B 82. Factor: B ÅBÄ" 83. Factor: "'B Å%!BÄ& 84. Factor: %*Ä&'CÄ"'C % % 85. Factor: > Ä!> Ä"!! 86. Factor: ; Å'; Ä* Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 9 of 15
10 cd i Recognizing Differences of Squares Then (Multiplying)... Now (Factoring)... a+ä, ba+å, b ú+ Å, + Å, ú a+ä, ba+å, b How to recognize a Difference of Squares. E There must be two expressions, both squares, ' ) such as %B fl*fl &> fl"fl Bfl%*C fi F They must have different signs. 87. Is *B Å'% a difference of squares? 88. Is B Å% a difference of squares? $ 89. Is &Å> a difference of squares? 90. Is B Ä& a difference of squares? cd j Factoring Differences of Squares 91. Factor: B Å% 92. Factor: C Å% 93. Factor: *Å"'> 94. Factor: % Å%*Ä> 95. Factor: 7 Å%: 96. Factor: &> Å7 " " 97. Factor: B Å 98. Factor: &Å B * %* Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 10 of 15
11 ck d Advanced Differences of Squares 99. Factor: ")B Å&!B 100. Factor: ' )B Å*) 101. Factor: %*B Å*B 102. Factor: % ' % %*+ Å)" 103. True or False?: + Å, ú a+å, b 104. True or False?: %+ Å*, ú a+å$, b % % 105. Factor completely: : Å"' 106. Factor: &B Å%!& % " " 107. Factor completely: C Å"'B 108. Factor: B Å"' Factoring Hints 1. Always look first for a common factor. If there is one, factor out the largest common factor. 2. Always factor completely. 3. Check by multiplying. Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 11 of 15
12 11.6 Solving Quadratic Equations by Factoring cd l Solving Quadratic Equations by Factoring Zero Factor Property If + and, are real numbers and if +, ú!, then +ú! or,ú! abå$ babä" b ú! 110. abå( babä b ú! 111. abå& babä( b ú! 112. abå"! ba$bä" b ú! 113. B&BÅ a b ú! 114. B%BÅ$ a b ú! 115. B Å*BÅ ú! 116. B Å$BÅ") ú! Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 12 of 15
13 117. B Å*B úå! 118. B Å"%B úå% cmd Solving Equations with Degree Greater than Two by factoring 119. $ $B Å"B ú! 120. $ B Å")B ú! Chapter 11: Factoring Polynomials Mr. Getso s Algebra Notes page 13 of 15
14 11.7 Quadratic Equations and Problem Solving 1. UNDERSTAND the problem. During this step, become comfortable with the problem. Some ways of doing this are: Read and re-read the problem. Construct a drawing. Propose a solution and check. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem. Choose a variable to represent the unknown. Use this variable to represent any other unknowns. 2. TRANSLATE the problem into an equation. 3. SOLVE the equation. 4. INTERPRET the results. Check the proposed solution in the stated problem and state your conclusion. Examples: 1. A window washer accidentally drops a bucket from the top of a 400-foot building. The height h of the bucket after t seconds is given by h = When will the bucket hit the ground? A. 5 sec 2. An object is thrown upward from the top of an 80-foot building with an initial velocity of 64 feet per second. The height h of the object after t seconds is given by the quadratic equation h = When will the object hit the ground? B. 25 sec C. 80 sec D. 5 sec
15 3. An object is thrown upward from the top of an 96-foot building with an initial velocity of 80 feet per second. The height h of the object after t seconds is given by the quadratic equation h = When will the object hit the ground? 4. The length of the base of a triangle is twice its height. If the area of the triangle is 225 square kilometers, find the height. 5. An object is dropped from 21 feet below the tip of the pinnacle atop a 697-ft tall building. The height h of the object after t seconds is given by the quadratic equation h = Find how many seconds pass before the object reaches the ground.
Instructor: Richard Getso Course: Math 200.P10 TR 1:00 PM Spring 2016 (Getso)
1/8/016 Practice Test 1 (Chapter 11) Richard Getso Student: Richard Getso Date: 1/8/16 Instructor: Richard Getso Course: Math 00.P10 TR 1:00 PM Spring 016 (Getso) Assignment: Practice Test 1 (Chapter 11)
Complex Numbers and Quadratic Equations
Chapter 16: Quadratic Equations 99 Chapter 16 Complex Numbers and Quadratic Equations Review: Simplifying Radicals The Product Rule for Radicals For any non negative radicands Eand F,» EÜ» Fú» EÜF Simplify
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
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
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:
Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents
Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations
Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.
Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
Slide 1 / 200. Quadratic Functions
Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
Mini-Lecture 5.1 Exponents and Scientific Notation
Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate
Section 1.7: Solving Equations by Factoring
Objective: Solve equations by factoring and using the zero product rule. When solving linear equations such as x 5 1, we can solve for the variable directly by adding 5 and dividing by to get 1. However,
Find a common monomial factor. = 2y 3 (y + 3)(y 3) Difference of two squares
EXAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 + 2x 2 15x = x(x 2 + 2x 15) Factor common monomial. = x(x + 5)(x 3) Factor trinomial. b. 2y 5 18y 3 = 2y 3 (y 2 9) Factor
Math 90 Final Exam Review ( )
Math 90 Final Eam Review (01 015) Course Learning Outcome 1: Model, interpret and justif mathematical ideas and concepts using multiple representations, and use multiple strands in mathematics, such as
Multiplying Monomials
320 Chapter 5 Polynomials Eample 1 Multiplying Monomials Multiply the monomials. a. 13 2 y 7 215 3 y2 b. 1 3 4 y 3 21 2 6 yz 8 2 a. 13 2 y 7 215 3 y2 13 521 2 3 21y 7 y2 15 5 y 8 Group coefficients and
Polynomials 370 UNIT 10 WORKING WITH POLYNOMIALS. The railcars are linked together.
UNIT 10 Working with Polynomials The railcars are linked together. 370 UNIT 10 WORKING WITH POLYNOMIALS Just as a train is built from linking railcars together, a polynomial is built by bringing terms
Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017
Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017 Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials
Unit 2 Quadratics. Mrs. Valentine Math 3
Unit 2 Quadratics Mrs. Valentine Math 3 2.1 Factoring and the Quadratic Formula Factoring ax 2 + bx + c when a = ±1 Reverse FOIL method Find factors of c that add up to b. Using the factors, write the
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
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
Algebra I Notes Unit Eleven: Polynomials
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials
Unit 5 Quadratic Expressions and Equations
Unit 5 Quadratic Expressions and Equations Test Date: Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials and
Quadratic Expressions and Equations
Unit 5 Quadratic Expressions and Equations 1/9/2017 2/8/2017 Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials
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
= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:
Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers
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,
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
SECTION 1.4 PolyNomiAls 4 1 learning ObjeCTIveS In this section, you will: Identify the degree and leading coefficient of polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL to multiply
Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.
LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in
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
INTERMEDIATE ALGEBRA REVIEW FOR TEST 3
INTERMEDIATE ALGEBRA REVIEW FOR TEST 3 Evaluate the epression. ) a) 73 (-4)2-44 d) 4-3 e) (-)0 f) -90 g) 23 2-4 h) (-2)4 80 i) (-2)5 (-2)-7 j) 5-6 k) 3-2 l) 5-2 Simplify the epression. Write your answer
5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
Practice Mixed Exercises. Write each polynomial in standard form. Then name each polynomial by its degree and the number of its terms.
Practice 10-1 Write each polynomial in standard form. Then name each polynomial by its degree and the number of its terms. 1. 4y 3 4y 2 3 y 2. x 2 x 4 6 3. x 2 4. 2m 2 7m 3 3m 5. 4 x 2x 2 6. 7x 3 2x 2
Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College
Lecture Guide Math 90 - Intermediate Algebra to accompany Intermediate Algebra, 3rd edition Miller, O'Neill, & Hyde Prepared by Stephen Toner Victor Valley College Last updated: 4/17/16 5.1 Exponents &
REVIEW SHEETS ELEMENTARY ALGEBRA MATH 65
REVIEW SHEETS ELEMENTARY ALGEBRA MATH 65 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets
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
When factoring, we ALWAYS start with the (unless it s 1).
Math 100 Elementary Algebra Sec 5.1: The Greatest Common Factor and Factor By Grouping (FBG) Recall: In the product XY, X and Y are factors. Defn In an expression, any factor that is common to each term
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
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
Math 2 Variable Manipulation Part 3 Polynomials A
Math 2 Variable Manipulation Part 3 Polynomials A 1 MATH 1 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not
Summer MA Lesson 11 Section 1.5 (part 1)
Summer MA 500 Lesson Section.5 (part ) The general form of a quadratic equation is a + b + c = 0, where a, b, and c are real numbers and a 0. This is a second degree equation. There are four ways to possibly
Name: 2. Determine (a) the leading coefficient and (b) the degree of the polynomials
Name: 1. For the following polynomial, 3x 3 x 2 + 4x 6 (a) List the terms: (b) List the coefficient of each term: (c) List the degree of each term: (d) Find the degree of the polynomial. 2. Determine (a)
Adding and Subtracting Polynomials
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
The greatest common factor, or GCF, is the largest factor that two or more terms share.
Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of
Unit 2, Ongoing Activity, Little Black Book of Algebra II Properties
Unit 2, Ongoing Activity, Little Black Book of Algebra II Properties Little Black Book of Algebra II Properties Unit 2 - Polynomial Equations & Inequalities 2.1 Laws of Exponents - record the rules for
Math 110 Final Exam Review Revised October 2018
Math 110 Final Exam Review Revised October 2018 Factor out the GCF from each polynomial. 1) 60x - 15 2) 7x 8 y + 42x 6 3) x 9 y 5 - x 9 y 4 + x 7 y 2 - x 6 y 2 Factor each four-term polynomial by grouping.
UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name:
UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay
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
Introductory Algebra Chapter 9 Review
Introductory Algebra Chapter 9 Review Objective [9.1a] Find the principal square roots and their opposites of the whole numbers from 0 2 to 2 2. The principal square root of a number n, denoted n,is the
The P/Q Mathematics Study Guide
The P/Q Mathematics Study Guide Copyright 007 by Lawrence Perez and Patrick Quigley All Rights Reserved Table of Contents Ch. Numerical Operations - Integers... - Fractions... - Proportion and Percent...
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
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
A. B. C. D. Quadratics Practice Test. Question 1. Select the graph of the quadratic function. g (x ) = 1 3 x 2. 3/8/2018 Print Assignment
Question 1. Select the graph of the quadratic function. g (x ) = 1 3 x 2 C. D. https://my.hrw.com/wwtb2/viewer/printall_vs23.html?umk5tfdnj31tcldd29v4nnzkclztk3w8q6wgvr262aca0a5fsymn1tfv8j1vs4qotwclvofjr8xhs0cldd29v4
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
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
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 +
Algebra I. Book 2. Powered by...
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..........
Math 110 Final Exam Review Revised December 2015
Math 110 Final Exam Review Revised December 2015 Factor out the GCF from each polynomial. 1) 60x - 15 2) 7x 8 y + 42x 6 3) x 9 y 5 - x 9 y 4 + x 7 y 2 - x 6 y 2 Factor each four-term polynomial by grouping.
Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to
Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to Learning Target #1: Factoring Factor the GCF out of a polynomial Factor a polynomial when a = 1 Factor
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
Just DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions :
5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #1-1 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull
degree -6x 3 + 5x 3 Coefficients:
Date P3 Polynomials and Factoring leading coefficient degree -6 3 + 5 3 constant term coefficients Degree: the largest sum of eponents in a term Polynomial: a n n + a n-1 n-1 + + a 1 + a 0 where a n 0
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
Algebra 2/Trig Apps: Chapter 5 Quadratics Packet
Algebra /Trig Apps: Chapter 5 Quadratics Packet In this unit we will: Determine what the parameters a, h, and k do in the vertex form of a quadratic equation Determine the properties (vertex, axis of symmetry,
ACTIVITY: Factoring Special Products. Work with a partner. Six different algebra tiles are shown below.
7.9 Factoring Special Products special products? How can you recognize and factor 1 ACTIVITY: Factoring Special Products Work with a partner. Six different algebra tiles are shown below. 1 1 x x x 2 x
Something that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.
Lesson Objectives: Students will be able to define, recognize and use the following terms in the context of polynomials: o Constant o Variable o Monomial o Binomial o Trinomial o Polynomial o Numerical
Prerequisites. Introduction CHAPTER OUTLINE
Prerequisites 1 Figure 1 Credit: Andreas Kambanls CHAPTER OUTLINE 1.1 Real Numbers: Algebra Essentials 1.2 Exponents and Scientific Notation 1.3 Radicals and Rational Expressions 1.4 Polynomials 1.5 Factoring
F.4 Solving Polynomial Equations and Applications of Factoring
section F4 243 F.4 Zero-Product Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or first-degree, equations. Solving higher
Chapter 1 Notes: Quadratic Functions
19 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form
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
Evaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply
Name EVALUATING ALGEBRAIC EXPRESSIONS Objective: To evaluate an algebraic expression Example Evaluate the expression if and y = 5 6x y Original problem 6() ( 5) Substitute the values given into the expression
5 Section 9.1 Prop of Radicals. 7 Section Section 9. 1b Properties of Radicals. 8 Quick Quiz Section 9.4 Completing the Square
Unit 10B Quadratics - Chapter 9 Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and make
B.3 Solving Equations Algebraically and Graphically
B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find
Algebra 1B Final Review
Name: Class: Date: ID: A Algebra 1B Final Review Short Answer 1. Originally a rectangle was twice as long as it was wide. When 5 m was subtracted from its length and 3 m subtracted from its width, the
1-3lab accuracy 6 module 1 Relationships between quantities ch1-1 variables and expressions 7,8 ch1-2 order of operations 7,8 ch 1-3 properties of
module 0 standards 1-3lab accuracy 6 module 1 Relationships between quantities ch1-1 variables and expressions 7,8 ch1-2 order of operations 7,8 ch 1-3 properties of numbers 7,8 1-3 lab accuracy 6 ch1-4
8 th Grade Honors Variable Manipulation Part 3 Student
8 th Grade Honors Variable Manipulation Part 3 Student 1 MULTIPLYING BINOMIALS-FOIL To multiply binomials, use FOIL: First, Outer, Inner, Last: Example: (x + 3)(x + 4) First multiply the First terms: x
Berkeley Math Circle, May
Berkeley Math Circle, May 1-7 2000 COMPLEX NUMBERS IN GEOMETRY ZVEZDELINA STANKOVA FRENKEL, MILLS COLLEGE 1. Let O be a point in the plane of ABC. Points A 1, B 1, C 1 are the images of A, B, C under symmetry
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
Equations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other
6.1 Quadratic Expressions, Rectangles, and Squares. 1. What does the word quadratic refer to? 2. What is the general quadratic expression?
Advanced Algebra Chapter 6 - Note Taking Guidelines Complete each Now try problem in your notes and work the problem 6.1 Quadratic Expressions, Rectangles, and Squares 1. What does the word quadratic refer
Mission 1 Factoring by Greatest Common Factor and Grouping
Algebra Honors Unit 3 Factoring Quadratics Name Quest Mission 1 Factoring by Greatest Common Factor and Grouping Review Questions 1. Simplify: i(6 4i) 3+3i A. 4i C. 60 + 3 i B. 8 3 + 4i D. 10 3 + 3 i.
Math 2 1. Lesson 4-5: Completing the Square. When a=1 in a perfect square trinomial, then. On your own: a. x 2 18x + = b.
Math 1 Lesson 4-5: Completing the Square Targets: I can identify and complete perfect square trinomials. I can solve quadratic equations by Completing the Square. When a=1 in a perfect square trinomial,
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
Geometry Summer Review Packet Page 1
June 017 Dear Geometry Students and Parents: Welcome to Geometry! For the 017-018 school year, we would like to focus your attention to the fact that many concepts from Algebra I are infused into Geometry.
Math 0312 EXAM 2 Review Questions
Name Decide whether the ordered pair is a solution of the given system. 1. 4x + y = 2 2x + 4y = -20 ; (2, -6) Solve the system by graphing. 2. x - y = 6 x + y = 16 Solve the system by substitution. If
Additional Exercises 10.1 Form I Solving Quadratic Equations by the Square Root Property
Additional Exercises 10.1 Form I Solving Quadratic Equations by the Square Root Property Solve the quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators.
Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1-10.1, not including 8.2) Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. 1. Factor completely: a 2
2 If ax + bx + c = 0, then x = b) What are the x-intercepts of the graph or the real roots of f(x)? Round to 4 decimal places.
Quadratic Formula - Key Background: So far in this course we have solved quadratic equations by the square root method and the factoring method. Each of these methods has its strengths and limitations.
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
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
Chapter 9 Quadratic Functions and Equations
Chapter 9 Quadratic Functions and Equations 1 9 1Quadratic Graphs and their properties U shaped graph such as the one at the right is called a parabola. A parabola can open upward or downward. A parabola
Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x
Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background
Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked
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
Solving Quadratic Equations
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
Applications Using Factoring Polynomials
Applications Using Factoring Polynomials This section will discuss applications involving the area of a rectangle, consecutive integers, and right triangles. Recall the steps that will help to translate
For Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c
9.5 Factor x2 1 bx 1 c Before You factored out the greatest common monomial factor. Now You will factor trinomials of the form x 2 1 bx 1 c. Why So you can find the dimensions of figures, as in Ex. 61.
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
How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials
PRC Ch P_3.notebook How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials How to remove common factors from polynomials
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.
Algebra 2 Honors Summer Review
Algebra Honors Summer Review 07-08 Label each problem and do all work on separate paper. All steps in your work must be shown in order to receive credit. No Calculators Allowed. Topic : Fractions A. Perform
Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)
Name: Per.: Date: Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Your Algebra 1 Final will be on at. You will need to bring your tetbook and number 2 pencils with you to the final eam.