Algebra. x + a 0. x 2 + a 1. are constants and a n. x n 1,..., a 0. x n, a n 1

Size: px
Start display at page:

Download "Algebra. x + a 0. x 2 + a 1. are constants and a n. x n 1,..., a 0. x n, a n 1"

Transcription

1 2 Algebra Introduction We have studied about a particular type of algebraic expression, called polynomial, and the terminology related to it. Now, we shall study the factorisation of polynomials. In addition, we shall study some more algebraic identities and their use in factorisation. Polynomials in one variable: A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0, where a 0, a 1, a 2,..., a n are constants and a n 0. Here, a 0, a 1, a 2,..., a n are respectively the coefficients of x 0, x, x 2,..., x n, and n (the highest power is called the degree of the polynomial p(x). Each of a n x n, a n 1 x n 1,..., a 0 with a n 0, is called a term of polynomial (px). Polynomials are classified according to the number of their terms as well as according to their degree. (i) A polynomial of one term is called a monomial. Examples: 5x 2, 4x, 54x 3, etc. (ii) A polynomial of two terms is called a binomial. Examples: x + 1, x 2 x, y 2 + 1, etc. (iii) A polynomial of three terms called a trinomial. Examples: x 2 + x + 2, 2 + x 2 x, etc. (iv) A polynomial of degree 0 is called a constant polynomial. Examples: 12, 74, 84, etc. (v) A polynomial of degree 1 is called a linear polynomial. Examples: 2x 2, 2y + 1, etc. (vi) A polynomial of degree 2 is called a quadratic polynomial. Examples: 2x 2 + 5, 5x 2 + 3x, etc. (vii) A polynomial of degree 3 is called a cubic polynomial. Examples: 3x 3, 2x 3 + 1, 5x 3 + x 2, etc. Algebraic identities: An algebraic identity is an algebraic equation that is true for all values of the variables occurring in it. Algebraic identities are used to factorise algebraic expressions and also in computations. Some algebraic identities are as follows: (i) (a + b) 2 = a 2 + 2ab + b 2 (ii) (a b) 2 = a 2 2ab + b 2 (iii) a 2 b 2 = (a + b)(a b) (iv) (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca (v) (a + b) 3 = a 3 + b 3 + 3a 2 b + 3ab 2 (vi) (a b) 3 = a 3 b 3 3ab(a b) (vii) a 3 + b 3 = (a + b)(a 2 ab + b 2 ) (viii) a 3 b 3 = (a b)(a 2 + ab + b 2 ) Multiplication of polynomials: To multiply two polynomials, multiply each term in one polynomial by each term in the other polynomial and add those answers together. Simplify if needed. Factorisation of polynomials: To factorise quadratic polynomials of the type ax 2 + bx + c, where a 0 and a, b, c are constants, we have to write b as the sum of two numbers whose product is ac. Activity Plus in Mathematics-9 25

2 Activity 2.1 Algebraic Identity : (a + b) 3 = a 3 + 3ab(a + b) + b 3 To compute (a + b) 3, we extend the identity (a + b) 2 = a 2 + 2ab + b 2 as follows: (a + b) 3 = (a + b)(a + b) 2 = (a + b)(a 2 + 2ab + b 2 ) = a(a 2 + 2ab + b 2 ) + b(a 2 + 2ab + b 2 ) = a 3 + 2a 2 b + ab 2 + a 2 b + 2ab 2 + b 3 = a 3 + 3a 2 b + 3ab 2 + b 3 = a 3 + 3ab(a + b) + b 3 Objective To verify the algebraic identity: (a + b) 3 = a 3 + 3ab(a + b) + b 3. Pre-requisite knowledge (i ) Concept of a cube and a cuboid (ii ) Volume of a cube = Side Side Side (iii ) Volume of a cuboid = Length Breadth Height Materials Required Cardboard Glazed paper of various colours Sheets of white paper A pair of scissors Scale, Cutter, Sketch pens Geometry box, Fevicol Procedure (i ) Make a cube of side a units as shown in the figure. Its volume is a 3. Wrap a red glazed paper on it. Fig. 1 (ii ) Make another cube of side b units as shown in the figure. Its volume is b 3. Wrap a black glazed paper on it. Fig. 2 (iii ) Make three cuboids each of dimensions (a) (b) (a + b) units. Wrap them with green glazed papers as shown in the figure. Volume of each cuboid is a b(a + b) cubic units. Fig Activity Plus in Mathematics-9

3 (iv ) Arrange the above two cubes and three cuboids in such a way that they altogether make a cube as shown in the figure. Fig. 4 (v ) A cubed binomial (sum) is equal to the cube of the first, plus three times the square of the first by the second, plus three times the first by the square of the second, plus the cube of the second. Observations (i ) We have joined the above five blocks (two cubes and three cuboids) to form a cube of side (a + b) units. (ii ) The volume of this cube is (a + b) 3 cubic units. (iii ) The volume of the two cubes are a 3 and b 3 cubic units. (iv ) The volume of each of the cuboid is ab(a + b) cubic units. i.e., the volume of all the three cuboids is 3ab(a + b) cubic units. Conclusion (a + b) 3 = a 3 + b 3 + 3[ab(a + b)] or (a + b) 3 = a 3 + 3ab(a + b) + b 3 Learning Outcomes (i ) The students will obtain the skill of adding the volumes of cubes and cuboids. (ii ) Showing the volume of a cube as the sum of cubes and cuboids helps the students to get a geometric feeling of volume. Remark The identity (a + b) 3 = a 3 + b 3 + 3ab(a + b) is useful for calculating the cube of a number which can be expressed as the sum of two convenient numbers. Note: The blocks used in this activity can be cut-out from soft-wood or can be made using cardboard or thermocol sheet. Activity Plus in Mathematics-9 27

4 Suggested Activity 1. To verify that (y + z) 3 = y 3 + z 3 + 3y 2 z + 3yz 2 by taking y = 14 and z = 4. Viva Voce Q1. In the activity of (a + b) 3, what do you mean by 3a 2 b, 3ab 2? Ans. 3a 2 b represents sets of cuboids with volumes a a b. And, 3ab 2 represents three sets of cuboids with volumes a b b. Q2. If one side of a cube is (a + 2b), then what is the volume of the cube? Ans. Volume of cube = (a + 2b) 3. Q3. What is the degree of (5x + 7y) 3? Ans. 3 Q4. What is the coefficient of z 2 in (3y +4z) 3? Ans. 144y. Q5. Is (a + b) 3 binomial? Ans. No. Q6. What is the value of (x + 1) 3? Ans. x 3 + 3x 2 + 3x + 1. Q7. What is an equation? Ans. A statement of equality is called an equation. Q8. What is the difference between an equation and an identity? Ans. An equation is not true for all values of the variable whereas an identity is true for every value of the variable in it. Q9. The L.H.S. of an identity is (a + b) 3. What is its R.H.S.? Ans. a 3 + 3a 2 b + 3ab 2 + b 3. Q10. What is the R.H.S. of the identity whose L.H.S. is (a + 2) 3? Ans. a 3 + 6a a + 8. qqq 28 Activity Plus in Mathematics-9

5 Activity 2.2 Algebraic Identity : (a b) 3 = a 3 3ab(a b) b 3 To compute (a b) 3, we extend the identity (a b) 2 = a 2 2ab + b 2 as follows: (a b) 3 = (a b)(a b) 2 = (a b)(a 2 2ab + b 2 ) = a(a 2 2ab + b 2 ) b(a 2 2ab + b 2 ) = a 3 2a 2 b + ab 2 a 2 b + 2ab 2 b 3 = a 3 3a 2 b + 3ab 2 b 3 = a 3 3ab(a b) b 3 Objective To verify the algebraic identity: (a b) 3 = a 3 3ab(a b) b 3. Pre-requisite Knowledge (i ) Volume of a cube = Side Side Side (ii ) Volume of a cuboid = Length Breadth Height Materials Required Cardboard Glazed paper of red, green and black colours Sketch pen, Geometry box Fevicol, Scale, Cutter, White papers A pair of scissors, Cello tape Procedure (i ) Make a cube of b units as shown in figure-1. Its volume is b 3 cubic units. Wrap a red glazed paper on it. (ii ) Make another cube of side (a b) units as shown in figure 2 and wrap a black glazed paper on it. Its volume is (a b) 3 cubic units. Fig. 1 Fig. 2 (iii ) Make three cuboids each of dimensions a, b and (a b) units as shown in figure 3. Volume of each cuboid is a b (a b) cubic units. Wrap them with green glazed paper. Fig. 3 Activity Plus in Mathematics-9 29

6 (iv ) Arrange the above five blocks (two cubes and three cuboids) in such a manner that they form a big cube as shown below: Fig. 4 Observations (i ) Each side of the resultant cube is a units. [ (a b) + b = a] (ii ) Volume of this cube = a 3 cubic units. (iii ) This cube is formed by joining the five blocks (2 cubes and 3 cuboids). (iv ) The volume of the two cubes are (a b) 3 cubic units and b 3 cubic units. (v ) The volume of the three cuboids taken together is 3ab(a b) cubic units. (vi ) From the cube of side a units, if we remove a cube of volume b 3 cubic units (wrapped by a red glazed paper) and the three cuboids each of volume ab(a b) cubic units (wrapped by a green glazed paper), then we are left with a cube of side (a b) units. Conclusion (a b) 3 + b 3 + 3[ab(a b)] = a 3 or (a b) 3 = a 3 3ab(a b) b 3. Learning Outcomes (i ) The students will obtain the skill of adding the volume of cubes and cuboids. (ii ) The students will obtain the skill of making a big cube using cuboids and cubes. (iii ) Showing the volume of a cube as the sum of cubes and cuboids helps them to get a geometric feeling of volume. Remark We call the right hand side expression, i.e., a 3 3ab(a b) b 3 the expanded form of the left hand side expression, i.e., (a b) Activity Plus in Mathematics-9

7 Suggested Activity 1. To verify that (y z) 3 = y 3 3y 2 z + 3yz 2 z 3 by taking y = 12 and z = 2. Viva Voce Q1. What is the maximum number of zeros that a cubic polynomial can have? Ans. Three. Q2. Expand (x 3y) 3. Ans. x 3 27y 3 9x 2 y + 27xy 2. Q3. For evaluating (999) 3, which formula we should use? Ans. We should use (a b) 3 = a 3 b 3 3ab(a b) by taking a = 1000 and b = 1. Q4. How would you expand p 3 q 3 in terms of (p q) 3? Ans. We know that (p q) 3 = p 3 q 3 3pq(p q) = p 3 q 3 3p 2 q + 3pq 2 p 3 q 3 = (p q) 3 + 3p 2 q 3pq 2. Q5. If one side of a cube is given by (a b), then what is the volume of the cube? Ans. Volume of cube (a b) 3. Q6. Write the coefficient of x 3 in (5x 3y) 3. Ans Q7. The L.H.S. of an identity is (a b) 3. What is its R.H.S.? Ans. a 3 3a 2 b + 3ab 2 b 3. Q8. What is the RHS of the identity whose L.H.S. is (a 1) 3? Ans. a 3 3a 2 + 3a 1. Q9. What is the short form of 8x y x 2 y + 54xy 2? Ans. 8x y x 2 y + 54xy 2 = (2x) 3 + (3y) 3 + 3(4x 2 )(3y) + 3(2x)(9y 2 ) = (2x) 3 + (3y) 3 + 3(2x) 2 (3y) + 3(2x)(3y) 2 = (2x + 3y) 3 qqq Activity Plus in Mathematics-9 31

ILLUSTRATIVE EXAMPLES

ILLUSTRATIVE EXAMPLES CHAPTER Points to Remember : POLYNOMIALS 7. A symbol having a fied numerical value is called a constant. For e.g. 9,,, etc.. A symbol which may take different numerical values is known as a variable. We

More information

POLYNOMIALS CHAPTER 2. (A) Main Concepts and Results

POLYNOMIALS CHAPTER 2. (A) Main Concepts and Results CHAPTER POLYNOMIALS (A) Main Concepts and Results Meaning of a Polynomial Degree of a polynomial Coefficients Monomials, Binomials etc. Constant, Linear, Quadratic Polynomials etc. Value of a polynomial

More information

Algebraic Expressions

Algebraic Expressions Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

Class IX Chapter 2 Polynomials Maths

Class IX Chapter 2 Polynomials Maths NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the

More information

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form

1. A polynomial p(x) in one variable x is an algebraic expression in x of the form POLYNOMIALS Important Points 1. A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = a nx n +a n-1x n-1 + a 2x 2 +a 1x 1 +a 0x 0 where a 0, a 1, a 2 a n are constants

More information

Algebraic Expressions and Identities

Algebraic Expressions and Identities ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 Algebraic Expressions and Identities CHAPTER 9 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or

More information

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

Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis

More information

Algebraic Expressions and Identities

Algebraic Expressions and Identities 9 Algebraic Epressions and Identities introduction In previous classes, you have studied the fundamental concepts of algebra, algebraic epressions and their addition and subtraction. In this chapter, we

More information

Factorisation CHAPTER Introduction

Factorisation CHAPTER Introduction FACTORISATION 217 Factorisation CHAPTER 14 14.1 Introduction 14.1.1 Factors of natural numbers You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write

More information

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

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

More information

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

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17 1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

More information

5.3. Polynomials and Polynomial Functions

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

More information

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

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

More information

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

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

More information

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

A-2. Polynomials and Factoring. Section A-2 1 A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring

More information

Factoring Trinomials of the Form ax 2 + bx + c, a 1

Factoring Trinomials of the Form ax 2 + bx + c, a 1 Factoring Trinomials of the Form ax 2 + bx + c, a 1 When trinomials factor, the resulting terms are binomials. To help establish a procedure for solving these types of equations look at the following patterns.

More information

ISSUED BY KENDRIYA VIDYALAYA - DOWNLOADED FROM Chapter - 2. (Polynomials)

ISSUED BY KENDRIYA VIDYALAYA - DOWNLOADED FROM   Chapter - 2. (Polynomials) Chapter - 2 (Polynomials) Key Concepts Constants : A symbol having a fixed numerical value is called a constant. Example : 7, 3, -2, 3/7, etc. are all constants. Variables : A symbol which may be assigned

More information

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

1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved. 1.3 Algebraic Expressions Copyright Cengage Learning. All rights reserved. Objectives Adding and Subtracting Polynomials Multiplying Algebraic Expressions Special Product Formulas Factoring Common Factors

More information

Algebraic Expressions

Algebraic Expressions ALGEBRAIC EXPRESSIONS 229 Algebraic Expressions Chapter 12 12.1 INTRODUCTION We have already come across simple algebraic expressions like x + 3, y 5, 4x + 5, 10y 5 and so on. In Class VI, we have seen

More information

27 Wyner Math 2 Spring 2019

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

More information

Algebra I Lesson 6 Monomials and Polynomials (Grades 9-12) Instruction 6-1 Multiplying Polynomials

Algebra I Lesson 6 Monomials and Polynomials (Grades 9-12) Instruction 6-1 Multiplying Polynomials In algebra, we deal with different types of expressions. Grouping them helps us to learn rules and concepts easily. One group of expressions is called polynomials. In a polynomial, the powers are whole

More information

Solving Quadratic Equations

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

More information

Factors of Polynomials Factoring For Experts

Factors of Polynomials Factoring For Experts Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product

More information

Mathwithsheppard.weebly.com

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

More information

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

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

More information

Unit 5 Evaluation. Multiple-Choice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number

Unit 5 Evaluation. Multiple-Choice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number Name I.D. Number Unit Evaluation Evaluation 0 Second Year Algebra (MTHH 039 09) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus, and other

More information

Algebra I Polynomials

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

More information

Maintaining Mathematical Proficiency

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

More information

POLYNOMIAL: A polynomial is a or the

POLYNOMIAL: A polynomial is a or the MONOMIALS: CC Math I Standards: Unit 6 POLYNOMIALS: INTRODUCTION EXAMPLES: A number 4 y a 1 x y A variable NON-EXAMPLES: Variable as an exponent A sum x x 3 The product of variables 5a The product of numbers

More information

Controlling the Population

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

More information

POLYNOMIAL EXPRESSIONS PART 1

POLYNOMIAL EXPRESSIONS PART 1 POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so

More information

Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2

Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2 Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Factor each expression. 1. 3x 6y 2. a 2 b 2 3(x 2y) (a + b)(a b) Find each product. 3. (x 1)(x + 3) 4. (a + 1)(a 2 + 1) x 2 + 2x 3 a 3 + a 2 +

More information

ANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B)

ANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B) ANSWERS CLASS: VIII TERM - 1 SUBJECT: Mathematics TOPIC: 1. Rational Numbers Exercise: 1(A) 1. Fill in the blanks: (i) -21/24 (ii) -4/7 < -4/11 (iii)16/19 (iv)11/13 and -11/13 (v) 0 2. Answer True or False:

More information

I CAN classify polynomials by degree and by the number of terms.

I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 13-1

More information

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

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY

SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY This summer assignment is designed to ensure that you are prepared for Algebra II/ Trigonometry. Nothing on this summer assignment is new. Everything is a

More information

3.0 INTRODUCTION 3.1 OBJECTIVES 3.2 SOLUTION OF QUADRATIC EQUATIONS. Structure

3.0 INTRODUCTION 3.1 OBJECTIVES 3.2 SOLUTION OF QUADRATIC EQUATIONS. Structure UNIT 3 EQUATIONS Equations Structure 3.0 Introduction 3.1 Objectives 3.2 Solution of Quadratic Equations 3.3 Quadratic Formula 3.4 Cubic and Bioquadratic Equations 3.5 Answers to Check Your Progress 3.6

More information

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 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

More information

Roots and Coefficients Polynomials Preliminary Maths Extension 1

Roots and Coefficients Polynomials Preliminary Maths Extension 1 Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p

More information

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

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

More information

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

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

More information

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x

More information

Algebra. Practice Pack

Algebra. Practice Pack Algebra Practice Pack WALCH PUBLISHING Table of Contents Unit 1: Algebra Basics Practice 1 What Are Negative and Positive Numbers?... 1 Practice 2 Larger and Smaller Numbers................ 2 Practice

More information

Solving Quadratic and Other Polynomial Equations

Solving Quadratic and Other Polynomial Equations Section 4.3 Solving Quadratic and Other Polynomial Equations TERMINOLOGY 4.3 Previously Used: This is a review of terms with which you should already be familiar. Formula New Terms to Learn: Discriminant

More information

Unit 3 Factors & Products

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

More information

TECHNIQUES IN FACTORISATION

TECHNIQUES IN FACTORISATION TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation

More information

ALGEBRAIC EXPRESSIONS AND POLYNOMIALS

ALGEBRAIC EXPRESSIONS AND POLYNOMIALS MODULE - ic Epressions and Polynomials ALGEBRAIC EXPRESSIONS AND POLYNOMIALS So far, you had been using arithmetical numbers, which included natural numbers, whole numbers, fractional numbers, etc. and

More information

CM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation

CM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation CM204: Computational Mathematics General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of simplifying algebraic expressions.

More information

Multiplying Monomials

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

More information

Algebra I. Polynomials.

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

More information

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

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

More information

Equations in Quadratic Form

Equations in Quadratic Form Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written

More information

CLASS IX MATHS CHAPTER REAL NUMBERS

CLASS IX MATHS CHAPTER REAL NUMBERS Previous knowledge question Ques. Define natural numbers? CLASS IX MATHS CHAPTER REAL NUMBERS counting numbers are known as natural numbers. Thus,,3,4,. etc. are natural numbers. Ques. Define whole numbers?

More information

How could you express algebraically, the total amount of money he earned for the three days?

How could you express algebraically, the total amount of money he earned for the three days? UNIT 4 POLYNOMIALS Math 11 Unit 4 Introduction p. 1 of 1 A. Algebraic Skills Unit 4 Polynomials Introduction Problem: Derrek has a part time job changing tires. He gets paid the same amount for each tire

More information

Unit 2, Ongoing Activity, Little Black Book of Algebra II Properties

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

More information

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

Quick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors. Lesson 11-3 Quick-and-Easy Factoring BIG IDEA Some polynomials can be factored into polynomials of lower degree; several processes are available to fi nd factors. Vocabulary factoring a polynomial factored

More information

Polynomial expression

Polynomial expression 1 Polynomial expression Polynomial expression A expression S(x) in one variable x is an algebraic expression in x term as Where an,an-1,,a,a0 are constant and real numbers and an is not equal to zero Some

More information

Higher Portfolio Quadratics and Polynomials

Higher Portfolio Quadratics and Polynomials Higher Portfolio Quadratics and Polynomials Higher 5. Quadratics and Polynomials Section A - Revision Section This section will help you revise previous learning which is required in this topic R1 I have

More information

Multiplication of Polynomials

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

More information

Algebra 1 Course Syllabus. Algebra 1, Part 1

Algebra 1 Course Syllabus. Algebra 1, Part 1 Course Description: Algebra 1 Course Syllabus In Algebra 1, students will study the foundations of algebra, including the understanding of variables, expressions, and working with real numbers to simplify

More information

Section 5.2 Polynomials, Sums, and Differences

Section 5.2 Polynomials, Sums, and Differences Department of Mathematics Grossmont College October 2, 2012 4.1 Systems of Linear Equations in Two Variables Learning Objectives: Give the degree of a polynomial Add and subract polynomials evaluate a

More information

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

We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials: R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)

More information

Lecture 27. Quadratic Formula

Lecture 27. Quadratic Formula Lecture 7 Quadratic Formula Goal: to solve any quadratic equation Quadratic formula 3 Plan Completing the square 5 Completing the square 6 Proving quadratic formula 7 Proving quadratic formula 8 Proving

More information

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

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

More information

VANA VANI MAT.HR. SEC.SCHOOL Std VIII MATHEMATICS

VANA VANI MAT.HR. SEC.SCHOOL Std VIII MATHEMATICS VANA VANI MAT.HR. SEC.SCHOOL Std VIII MATHEMATICS Holiday assignment 1.Find the common factors of the terms (i) 12x, 36 (ii) 2y, 22xy (iii) 14pq, 28p 2 q 2 (iv) 2x, 3x 2, 4 (v) 6abc, 24ab 2, 12a 2 b (vi)

More information

Find two positive factors of 24 whose sum is 10. Make an organized list.

Find two positive factors of 24 whose sum is 10. Make an organized list. 9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is

More information

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

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 +

More information

Summer Prep Packet for students entering Algebra 2

Summer Prep Packet for students entering Algebra 2 Summer Prep Packet for students entering Algebra The following skills and concepts included in this packet are vital for your success in Algebra. The Mt. Hebron Math Department encourages all students

More information

( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.

( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3. Chapter 6 Exercise Set 6.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.

More information

Teacher's Page. Mar 20 8:26 AM

Teacher's Page. Mar 20 8:26 AM Teacher's Page Unit 4.1 in two parts: Part 1: Polynomials and Part 2: Quadratics Benchmarks: A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity

More information

Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 1

Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 1 Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 1 Algebraic Notation The ability to convert worded sentences and problems into algebraic symbols

More information

Polynomials and Factoring

Polynomials 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 information

Sections 7.2, 7.3, 4.1

Sections 7.2, 7.3, 4.1 Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals

More information

THE RING OF POLYNOMIALS. Special Products and Factoring

THE RING OF POLYNOMIALS. Special Products and Factoring THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products - rules

More information

Stepping Stones for Introduction to Algebra

Stepping Stones for Introduction to Algebra Quality for Equality Stepping Stones for Introduction to Algebra Section I - Introduction to Algebra 1) Play Think of a number game Think of a number ( single digit number between 1 and 9 ) Add 3 Multiply

More information

Vocabulary. Term Page Definition Clarifying Example. binomial. cubic. degree of a monomial. degree of a polynomial

Vocabulary. Term Page Definition Clarifying Example. binomial. cubic. degree of a monomial. degree of a polynomial CHAPTER 7 Vocabulary This table contains important vocabulary terms from Chapter 7. As you work through the chapter, fill in the page number, definition, and a clarifying example for each term. binomial

More information

West Windsor-Plainsboro Regional School District Math A&E Grade 7

West Windsor-Plainsboro Regional School District Math A&E Grade 7 West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale

More information

Ch 7 Summary - POLYNOMIAL FUNCTIONS

Ch 7 Summary - POLYNOMIAL FUNCTIONS Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)

More information

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

When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules

More information

POLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1

POLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1 POLYNOMIALS A polynomial in x is an expression of the form p(x) = a 0 + a 1 x + a x +. + a n x n Where a 0, a 1, a. a n are real numbers and n is a non-negative integer and a n 0. A polynomial having only

More information

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II Course Number 5116 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra 1 or Algebra 1

More information

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

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

More information

LESSON 7.2 FACTORING POLYNOMIALS II

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

More information

Math Lecture 18 Notes

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

More information

8.5 Addition and Subtraction of Polynomials

8.5 Addition and Subtraction of Polynomials 8.5. Addition and Subtraction of Polynomials www.ck12.org 8.5 Addition and Subtraction of Polynomials Learning Objectives Write a polynomial expression in standard form. Classify polynomial expression

More information

Question: 1. Use suitable identities to find the following products:

Question: 1. Use suitable identities to find the following products: CH-2 Polynomial Question: 1. Use suitable identities to find the following products: (i) (x + 4) (x + 10) Solution:- (x+4)(x+10) = x 2 +10x+4x+4 x 10 = x 2 +14x+40 (ii) (x + 8) (x 10) Solution: x 2-10x+8x-80

More information

2, or x 5, 3 x 0, x 2

2, or x 5, 3 x 0, x 2 Pre-AP Algebra 2 Lesson 2 End Behavior and Polynomial Inequalities Objectives: Students will be able to: use a number line model to sketch polynomials that have repeated roots. use a number line model

More information

AFM Review Test Review

AFM Review Test Review Name: Class: Date: AFM Review Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. What are the solutions of the inequality?. q + (q ) > 0 q < 3 q

More information

1. ALGEBRAIC EXPRESSIONS

1. ALGEBRAIC EXPRESSIONS Colegio Herma. Maths. Bilingual Departament by Isabel Martos Martínez. 2015 1. ALGEBRAIC EXPRESSIONS You can describe everyday situations by using algebra. In algebra, you use letters to represent unknown

More information

( 4 p 3. ( 2 p 2. ( x 3 y 4. ( y. (2 p 2 ) 2 ( q 4 ) 2. ( x 2 ) POLYNOMIALS, PAGES CHECK IT OUT! PAGES

( 4 p 3. ( 2 p 2. ( x 3 y 4. ( y. (2 p 2 ) 2 ( q 4 ) 2. ( x 2 ) POLYNOMIALS, PAGES CHECK IT OUT! PAGES 8. _ x 4 y 8 x 4-6 y 8-6 x 6 y 6 x - y y x 9. 5 m n 4 5 m - n 4-1 m n 5 m 0 n 3 30. ( 3 5) 3 3 3 31. _ ( 4 p 3 5 1 n 3 5 n 3 5 3 _ 7 15 4) p q ( 4 p 3-1 q -4 ) ( p q -4 ) ( p q 4 ) ( p ) ( q 4 ) _ ( p

More information

Lesson 3: Polynomials and Exponents, Part 1

Lesson 3: Polynomials and Exponents, Part 1 Lesson 2: Introduction to Variables Assessment Lesson 3: Polynomials and Exponents, Part 1 When working with algebraic expressions, variables raised to a power play a major role. In this lesson, we look

More information

Common Core Algebra 2 Review Session 1

Common Core Algebra 2 Review Session 1 Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x 2 + 2 (4) 12x

More information

Lecture 26. Quadratic Equations

Lecture 26. Quadratic Equations Lecture 26 Quadratic Equations Quadratic polynomials....................................................... 2 Quadratic polynomials....................................................... 3 Quadratic equations

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:

More information

Jakarta International School 8 th Grade AG1 Practice Test - BLACK

Jakarta International School 8 th Grade AG1 Practice Test - BLACK Jakarta International School 8 th Grade AG1 Practice Test - BLACK Polynomials and Quadratic Equations Name: Date: Grade: Standard Level Learning Goals - Green Understand and Operate with Polynomials Graph

More information

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

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,

More information

Gaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students

Gaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students Gaithersburg High School Math Packet For Rising Algebra 2/Honors Algebra 2 Students 1 This packet is an optional review of the skills that will help you be successful in Algebra 2 in the fall. By completing

More information

Collecting Like Terms

Collecting Like Terms MPM1D Unit 2: Algebra Lesson 5 Learning goal: how to simplify algebraic expressions by collecting like terms. Date: Collecting Like Terms WARM-UP Example 1: Simplify each expression using exponent laws.

More information

Polynomials and Factoring

Polynomials and Factoring Polynomials and Factoring Jeffrey Shi January 26, 2016 1 Polynomials 1.1 Definition Definition 1 (Polynomial) A polynomial is an algebraic expression involving one or more terms with one or more unknown

More information

MHF4U Unit 2 Polynomial Equation and Inequalities

MHF4U Unit 2 Polynomial Equation and Inequalities MHF4U Unit 2 Polynomial Equation and Inequalities Section Pages Questions Prereq Skills 82-83 # 1ac, 2ace, 3adf, 4, 5, 6ace, 7ac, 8ace, 9ac 2.1 91 93 #1, 2, 3bdf, 4ac, 5, 6, 7ab, 8c, 9ad, 10, 12, 15a,

More information