Unit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2


 Ernest Baker
 2 years ago
 Views:
Transcription
1 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with an equal sign. Eample: 5 Terms:  are separated by an addition or subtraction sign.  each term begins with the sign preceding the variable or coefficient. Numerical Coefficient Monomial:  one term epression. Binomial:  two terms epression. Eample: 5 Eample: 5 5 Eponent Variable Trinomial:  three terms epression. Eample: 5 Polynomial:  many terms (more than one) epression. All Polynomials must have whole numbers as eponents!! Eample: 9 is NOT a polynomial. Degree:  the term of a polynomial that contains the largest sum of eponents Eample: 9 y 5 y Degree (5 ) Eample : Fill in the table below. Polynomial Number of Terms Classification Degree Classified by Degree 9 monomial 0 constant monomial linear 9 binomial linear trinomial quadratic 9 polynomial cubic 9 trinomial quartic Like Terms:  terms that have the same variables and eponents. Eamples: y and 5 y are like terms y and 5y are NOT like terms Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.
2 Unit : Polynomials Pure Math 0 Notes To Add and Subtract Polynomials: Combine like terms by adding or subtracting their numerical coefficients. Eample : Simplify the followings. a. 5 b. (9 y y ) ( y 0 y ) 5 9 y y y 0 y 9 y y c. (9 y y ) ( y 0 y ) 9 y y y 0 y (drop brackets and switch signs in the bracket that had sign in front of it) 9 y y d. Subtract 9 5 This is the same as (9 ) (5 ) 9 5 To Multiply and Divide Monomials: Multiply or Divide (Reduce) Numerical Coefficients. Add or Subtract eponents of the same variable according to basic eponential laws. Eample : Simplify the followings. a. ( y ) ( y y z ) b. 5 yz ()() ( )( ) (y )(y 5 y z ) 5 y z 5 5a b c. 5 5a b 5 a b 5 5 a b 5 y y z 0 ( z 0 ) a b or b a y Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
3 Pure Math 0 Notes Unit : Polynomials (AP) Eample : Find the area of the following ring. General Formula for Area of a Circle A πr Inner Circle Radius Outer Circle Radius ( ) Inner Circle Area: A π () A π ( ) A π Outer Circle Area: A π () Shaded Area π π Shaded Area π A π ( ) A π  Homework Assignment Regular: pg. 00 # to 5, 55, 5 AP: pg. 00 # to 5, 55 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.
4 Unit : Polynomials Pure Math 0 Notes : Multiplying Polynomials To Multiply Monomials with Polynomials Eample : Simplify the followings. a. ( ) b. ( ) ( ) ( ) c. (5 ) ( ) d. (a a ) (a ) (only multiply (5 ) ( ) the brackets (a a ) (a ) right after the 5 monomial) a a a 5a a To Multiply Polynomials with Polynomials Eample : Simplify the followings. a. ( ) ( ) b. ( ) ( 5 ) ( ) ( ) ( ) ( 5 ) c. ( ) ( ) ( ) ( ) d. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 5 ) Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
5 Pure Math 0 Notes Unit : Polynomials Eample : Find the shaded area of each of the followings. a. 5 b. Shaded Area Big Rectangle Small Square (5 ) ( ) ( ) ( ) 5 (0 5 ) ( ) (0 ) ( ) 0 Shaded Area ( ) ( ) Total Area Top Rectangle Bottom Rectangle ( ) ( ) ( ) ( 5) ( ) ( 0 5) ( ) ( 9 5) 9 5 Total Area Homework Assignment Regular: pg # to (odd),, AP: pg # to (even), 5,,, 9 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 5.
6 Unit : Polynomials Pure Math 0 Notes : Special Products ( y) ( y) ( y) ( y) is NOT y ( y) ( y) ( y) ( y) Eample : Simplify the followings. a. ( ) b. ( ) ( ) ( ) ( ) ( ) ( ) 9 ( ) ( ) 9 ( ) ( ) c. ( ) ( ) Eample : Find the volume of the bo below. ( ) ( ) ( ) ( ) (9 ) ( ) (9 ) ( ) 9 5 Volume Length Width Height  Homework Assignment Regular: pg. # to (odd), 9, 5, 5 (a, c, e, g), 55 (a, c, e, g), 5 AP: pg. # to (even), 9 to 5, 5 (b, d, f, h), 55 (b, d, f, h), 5, 5 V ( ) ( ) V (9 ) ( ) V (9 ) ( ) V 9 Volume Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
7 Pure Math 0 Notes Unit : Polynomials : Common Factors Common Factors can consist of two parts: a. Numerical GCF:  Greatest Common Factor of all numerical coefficients and constant. b. Variable GCF:  the lowest eponent of a particular variable. After obtaining the GCF, use it to divide each term of the polynomial for the remaining factor. Eample : Factor the followings a. b. a b ab ab GCF ( ) ab (a b ) GCF ab Factor by Grouping (Common Brackets as GCF) a (c d) b (c d) (c d) (a b) Common Brackets Take common bracket out as GCF Eample : Factor. a. ( ) ( ) b. ab ac b bc ( ) ( ) (ab ac) (b bc) a (b c) b (b c) c. y 9 y (b c) (a b) ( y ) (9 y ) ( y ) (9 y ) Brackets are NOT the same! We might have to first rearrange terms. Try again after rearranging terms! 9 y y ( 9) (y y ) ( ) y ( ) Switch Sign in Second Bracket! We have put a minus sign in front of a new bracket! ( ) ( y ) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.
8 Unit : Polynomials Pure Math 0 Notes Eample : Find the area of the shaded region in factored form and as a polynomial. Shaded Area Area of Square Area of Circle Shaded Area π Area of a Circle A πr Shaded Area π (Polynomial Form) π Shaded Area (Factored Form) Radius of Circle  Homework Assignments Regular: pg. 0 # to 5 (odd), to AP: pg.0 # to (even), to Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
9 Pure Math 0 Notes Unit : Polynomials : Factoring b c (Leading Coefficient is ) b c What two numbers multiply to give c, but add up to be b? Eample : Completely factor the followings. a. 5 Product of b. 0 ( ) ( ) ( ) ( 5) sum of 5 Product of sum of c. a a 5 Product of d. y y (a ) (a 5) 5 5 ( y) ( y) sum of e. y y Product of f. 5w w (y ) (y ) w 5w sum of (w 5w ) (w ) (w ) g. ab ab 0a Product of sum of Rearrange in Descending Degree. Take out as common factor. () () () () 5 a (b b 0) a (b ) (b 5) Take out GCF () (5) 0 () (5) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 9.
10 Unit : Polynomials Pure Math 0 Notes Eample : List all values of k such that the trinomial k can be factored. Product of Possible Sums for k Eample : A rectangular has an area of 9 0. a. What are the dimensions of the rectangle? b. If 5 cm, what are the actual dimensions? Factor Area for dimensions Area length width if 5 cm Area 9 0 Area 9 0 Dimensions (5 0) (5 ) Dimensions ( 0) ( ) Dimensions 5 cm cm (AP) Eample : Factor the followings. a. b. ( ) ( ) ( ) ( ) y y Assume b c as the same as b c and factor. The answer will be ( ) ( ). (y ) (y ) ( ) ( ) Do NOT Epand!! Let y ( ) ( ) ( 5)  Homework Assignments Regular: pg. #9 to 59 (odd),, 5, AP: pg. #0 to 0 (even),, 5 Page 0. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
11 Pure Math 0 Notes Unit : Polynomials 9: Factoring a b c (Leading Coefficient is not, a ) For factoring trinomial with the form a b c, we will have to factor by grouping. Eample : Factor First, we look for GCF. But there is no GCF! Multiply a and c. Product of sum of Split the b term into two separate terms. ( ) ( ) Group by brackets ( ) ( ) Take out GCF for each bracket. ( ) ( ) Factor by Common Bracket! Eample : Completely factor the followings. () (9) a. y y 9 b. d d 9 9 y y y 9 d d d (y y) (y 9) (d d) (d ) sum of y (y ) (y ) d (d ) (d ) switch sign! (y ) (y ) ( sign in front (d ) (d ) of brackets) c. GCF d. m mn 9n 9 () () () () ( ) m mn mn 9n () () ( ) (m mn) (mn 9n ) [ ( sum of ) ( ) ] m (m n) n (m n) [ ( ) ( ) ] switch sign! (m n) (m n) ( sign in front ( ) ( ) of brackets) () () sum of switch sign! ( sign in front of brackets) switch sign! ( sign in front of brackets) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.
12 Unit : Polynomials Pure Math 0 Notes Eample : List all possible values for k in 5 k so it could be factored. Product of (5) () 0 Possible Sums for k (AP) Eample : Factor the followings. a. 9 9 b. y y 9 9 () (9) () (9) y y y ( ) (9 9) ( y) ( y y ) () () () () ( ) 9 ( ) ( y) y ( y) ( ) ( 9) ( y) ( y) switch sign! ( sign in front of brackets) 9 Homework Assignments Regular: pg. 0  # to 9 (odd), 5, 5, 55 AP: pg. 0  # to 50 (even), 5, 55 Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
13 Pure Math 0 Notes Unit : Polynomials 0: Factoring Special Quadratics Difference of Squares (Square Square) y ( y) ( y) Eample : Completely factor the followings. a. 5 b. 9 c. 00 ( 5) ( 5) (NOT Factorable Sum of Squares) ( 00) ( 0) ( 0) d. e. 9 y ( 9) ( 9) ( y) ( y) ( ) ( ) ( 9) (AP) Eample : Completely factor the followings. Look at ( ) and ( ) a. ( ) 9 Look at ( ) b. ( ) ( ) as individual items! as a single item! [ ( ) ] [ ( ) ] [ ( ) ( ) ] [ ( ) ( ) ] ( ) ( ) [ ] [5 ] ( ) (5 ) Perfect Trinomial Square a b c ( a c ) where a, c are square numbers, and b ( a )( c ) Watch Out! Subtracting a bracket! Take out negative sign from the first bracket! Eample : Epand ( ). ( ) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.
14 Unit : Polynomials Pure Math 0 Notes Eample : Completely factor the followings. a b ( 5) ( ) (AP) Eample 5: Factor Assumes b c is the same as b c. But the answer will be in the form of ( ) ( ). ( )( 00) ( 0) Eample : List all possible values for k that can make the following polynomials as perfect squares. a. k b. k 0 5 k k 0 5 k 0 k k and ()() c. 9 5y ky The middle term of any perfect trinomial squares can have a positive or a negative numerical coefficient! 0 0 k k ( k ) 9 5y ky 9 5 ( )( k ) 5 k k Page. ( k ) 0 Homework Assignments Regular: pg.  # to (odd), 5 to 5 AP: pg.  # to (even), to 5, 59,, Copyrighted by Gabriel Tang, B.Ed., B.Sc.
15 Pure Math 0 Notes A: Dividing Polynomials Consider 5. 5 R Original Number We can say that 5 Unit : Polynomials Divisor Quotient Divisor Original Number Divisor Quotient Or, we can say 5 () () Remainder Remainder Polynomial Function Divisor Function In general, for P() D(), we can write P( ) R Q( ) or P() D() Q() R D( ) D( ) Restriction: D() 0 Quotient Function Remainder NonPermissible Value (NPV):  restriction on what the variable CANNOT be equal to due to the fact that the Denominator CANNOT be 0. (You can never divide by 0!) Division with Monomials Eample : Simplify the followings y a. b. y 5 ( )( ) c NPV (NPV 0) (NPV 0) Divide each term of the polynomial by the monomial. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 5.
16 Unit : Polynomials Pure Math 0 Notes Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Long Division to Divide Polynomials Eample : Divide R Eample : Divide R You cannot divide monomial by polynomial! Dividing by Polynomial is only possible by Long Division! OR 9 5 ( ) ( ) 5 For NPV, we let 0 NPV: OR 5 ( ) ( 9) 0 For NPV, we let 0 NPV:
17 Pure Math 0 Notes Unit : Polynomials Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page. Eample : Divide 9 0 R Eample 5: Divide R 9 OR ( ) ( ) 9 For NPV, we let 0 NPV: 0 Missing Term from Decreasing Degree! OR ( ) ( ) ( ) For NPV, 0 No NPV! (cannot take square root of negative number) Missing Term from Decreasing Degree! 0 A Homework Assignments Regular: pg. 55 # to (odd), 5 (a, c, e),, 0 AP: pg. 55 # to (even), 5 (a, c, e),,,, 5,
18 Unit : Polynomials Pure Math 0 Notes B: Synthetic Division Only works well on divisor that is in a form of a, where Leading coefficient of Divisor is on the divisor. Eample : Divide Divisor Coefficients and constant of Polynomial Subtract numbers in each column Remainder 0 Quotient 9 (one degree less than the original polynomial) Eample : Divide 0 Quotient Remainder 9 9 Eample : Divide 5 5 Quotient 9 Remainder 9 Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.
19 Pure Math 0 Notes Unit : Polynomials The Remainder Theorem If you want to find only the remainder, you can simply substitute a from the Divisor, ( a), into the original Polynomial, P(). In general, when P( ) a, P(a) Remainder Eample : Find the remainder of the followings. a. 5 b. 0 0 P() () () 5() P() () () 0 5 Remainder 0 Remainder 9 c. 5 0 P() () () 5() 0 Remainder B Homework Assignments Regular: pg. 5 # to ; pg. 55 Section #5 (a, b, c) AP: pg. 5 # to ; pg. 55 Section #5 (a, b, c) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 9.
PreCalculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
More informationDIVISION OF POLYNOMIALS
5.5 Division of Polynomials (533) 89 5.5 DIVISION OF POLYNOMIALS In this section Dividing a Polynomial by a Monomial Dividing a Polynomial by a Binomial Synthetic Division Division and Factoring We began
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 informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
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 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 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 information3. Power of a Product: Separate letters, distribute to the exponents and the bases
Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same
More informationPlacement Test Review Materials for
Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the
More informationMonomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ.
Chapter 7.1 Introduction to Polynomials A monomial is an expression that is a number, a variable or the product of a number and one or more variables with nonnegative exponents. Monomials that are real
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
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 informationx n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.
Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense
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 informationSimplifying Exponential Expressions
Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write
More informationA.1 Radicals and Rational Exponents
APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationDividing Polynomials VOCABULARY
 Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics
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 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 informationGreatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
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 informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
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 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 informationUnit: Polynomials and Factoring
Name Unit: Polynomials: Multiplying and Factoring Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least common multiple
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 informationWhen factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.
Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
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 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 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 informationAlgebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials
Algebra Unit 6 Syllabus revised /7/13 1 Objective: Multiply monomials. Simplify expressions involving powers of monomials. Preassessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages
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 information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationDividing Polynomials; Remainder and Factor Theorems
Dividing Polynomials; Remainder and Factor Theorems In this section we will learn how to divide polynomials, an important tool needed in factoring them. This will begin our algebraic study of polynomials.
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
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 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 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 informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationChapter 3 Section 6 Lesson Polynomials
Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.
More information78 Multiplying Polynomials
78 Multiplying Polynomials California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these
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 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:
Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?
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 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 informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationCLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.
SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
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 informationFlorida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies  Lower and Upper
Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies  Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,
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 informationREVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95
REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 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
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 informationPreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to:
PreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGrawHill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button
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 informationState whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence.
State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence. 1. x + 5x + 6 is an example of a prime polynomial. The statement is false. A
More informationMAT 0950 Course Objectives
MAT 0950 Course Objectives 5/15/20134/27/2009 A student should be able to R1. Do long division. R2. Divide by multiples of 10. R3. Use multiplication to check quotients. 1. Identify whole numbers. 2. Identify
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : NLYTICL METHODS FOR ENGINEERS Unit code: /0/0 QCF Level: Credit value: OUTCOME TUTORIL LGEBRIC METHODS Unit content Be able to analyse and model engineering situations and solve problems using algebraic
More informationAn Insight into Division Algorithm, Remainder and Factor Theorem
An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the
More informationP.E.R.T. Math Study Guide
A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT  A Math Study Guide 1. Linear Equations
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 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 informationOperations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationSection A3 Polynomials: Factoring APPLICATIONS. A22 Appendix A A BASIC ALGEBRA REVIEW
A Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) 4 54. ( ) 3 ( ) 3( ) 7 55. 3{[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by
More information52 Dividing Polynomials. Simplify. 2. (3a 2 b 6ab + 5ab 2 )(ab) 1 SOLUTION: 4. (2a 2 4a 8) (a + 1) SOLUTION: 6. (y 5 3y 2 20) (y 2) SOLUTION:
Simplify. 2. (3a 2 b 6ab + 5ab 2 )(ab) 1 4. (2a 2 4a 8) (a + 1) 6. (y 5 3y 2 20) (y 2) esolutions Manual  Powered by Cognero Page 1 Simplify. 8. (10x 2 + 15x + 20) (5x + 5) 10. 12. Simplify esolutions
More informationPolynomial Expression
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationSimplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
More information6.4 Factoring Polynomials
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
More information83 Multiplying Polynomials. Find each product. 1. (x + 5)(x + 2) SOLUTION: 2. (y 2)(y + 4) SOLUTION: 3. (b 7)(b + 3) SOLUTION:
Find each product. 1. (x + 5)(x + ). (y )(y + 4) 3. (b 7)(b + 3) 4. (4n + 3)(n + 9) 5. (8h 1)(h 3) 6. (a + 9)(5a 6) 7. FRAME Hugo is designing a frame as shown. The frame has a width of x inches all the
More informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationFactoring Algebra Chapter 8B Assignment Sheet
Name: Factoring Algebra Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.
More informationExponents. Learning Objectives 41
Eponents 1 to  Learning Objectives 1 The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient We can simplify by combining the like
More informationLESSON 6.2 POLYNOMIAL OPERATIONS I
LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order
More informationQuadratic Equations and Applications
77 Quadratic Equations and Applications 38 77 Quadratic Equations and Applications Quadratic Equations Recall that a linear equation is one that can be written in the form a b c, where a, b, and c are
More informationMATHCOUNTS TOOLBOX Facts, Formulas and Tricks
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.
More informationFactoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
More informationPark Forest Math Team. Meet #5. Algebra. Selfstudy Packet
Park Forest Math Team Meet #5 Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More informationUNIT TWO POLYNOMIALS MATH 421A 22 HOURS. Revised May 2, 00
UNIT TWO POLYNOMIALS MATH 421A 22 HOURS Revised May 2, 00 38 UNIT 2: POLYNOMIALS Previous Knowledge: With the implementation of APEF Mathematics at the intermediate level, students should be able to: 
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
More informationExponents and Polynomials
Because of permissions issues, some material (e.g., photographs) has been removed from this chapter, though reference to it may occur in the tet. The omitted content was intentionally deleted and is not
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 informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
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 informationDeriving the Equation of a Circle. Adapted from Walch Education
Deriving the Equation of a Adapted from Walch Education s A circle is the set of all points in a plane that are equidistant from a reference point in that plane, called the center. The set of points forms
More informationMath 1111 Journal Entries Unit I (Sections , )
Math 1111 Journal Entries Unit I (Sections 1.11.2, 1.41.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
More information