MATHEMATICS CLASS IX

Jishnusagar V S S III B.Ed. Mathematics Karmela Rani Training College Kollam 1

NATIONAL ANTHEM Jana Gana Mana Adhinayaka Jay He Bharata Bhagya Vidhata Panjab Sindhu Gujarat Maratha Dravida Utkala Banga Vindhya Himachal Yamuna Ganga Uchchala Jaladhi Taranga Tava Subha Name Jaage Tave Subha Aashish Mange Gaahe Tava Jay Gaatha Jana Gana Mangal Daayak Jay He Bharat Bhagya Vidhata Jay he Jay he Jay he Jay Jay Jay Jay he

PLEDGE India is my country and all Indians are my brothers and sisters. I love my country and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give respect to my parents, teachers and elders and treat everyone with courtesy.to my country and my people, I pledge my devotion. In their wellbeing and prosperity alone, lies my happiness.

2

PREAMBLE OF INDIAN CONSTITUTION

3

Dear students, Starting your day with a positive attitude towards

learning and your life of study can be an enormous help to make your day in school go smoothly. Just a few minutes of reflection and writing down one positive thought in a gratitude journal every day will gradually form a more positive outlook. You might not see the effects immediately but if you tend to worry, try it and see the difference it can make to your day. May your journey through this book make learning a joyful experience. With love and regard, Jishnusagar V S

4

CHAPTER 1

POLYNOMIALS 1.1

Introduction

1.2

Polynomials in one variable

1.3

Polynomials โ Related Terminologies

1.4

Degree of a polynomial

5

POLYNOMIALS

1.1 INTRODUCTION

What is a polynomial The word polynomial is derived from the Greek words poly(meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms". Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.

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1.2 POLYNOMIALS IN ONE VARIABLE Polynomials in one variable is an algebraic expression that consists of one variable in it. Some of the examples of polynomials in one variable are given below

โข

๐ฅ+2

โข

๐ฅยฒ + 3๐ฅ + 2

โข

๐ฆยณ + 2๐ฆยฒ โ ๐ฆ + 5

Recall that a variable is denoted by a symbol that can take any real value. We use the letters ๐ฅ, ๐ฆ, ๐ง, etc. to denote variables. 2๐ฅ, โ ๐ฅ, โ 1/2 ๐ฅ are all algebraic expressions These expressions are of the form (a constant) ร ๐ฅ. Now suppose we want to write an expression which is (a constant) ร (a variable) and we do not know what the constant is. In such cases, we write the constant as ๐, ๐, ๐, etc. So, the expression will be ๐๐ฅ, say.

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Fig 1.1 2

2 Consider a square of side 2 units (see Fig. 1.1). What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its four sides. Here, each side is 2 units. So, its perimeter is 4 ร 2, i.e., 8 units. What will be the perimeter if each side of the square is 10 units? The perimeter is 4 ร 10, i.e., 40 units. In case the length of each side is x units (see Fig. 2.2), the perimeter is given by 4๐ฅ units. So, as the length of the side varies, the perimeter varies. Fig 1.2

๐ฅ 8

If you find the area of the square PQRS? It is ๐ฅ ร ๐ฅ = ๐ฅ 2 square units. = ๐ฅ 2 is an algebraic expression. You are also familiar with other algebraic expressions like 2๐ฅ, ๐ฅยฒ + 2๐ฅ, ๐ฅยณโ ๐ฅยฒ + 4๐ฅ + 7. Note that, all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable. Expressions of this form are called polynomials in one variable. For instance, 2๐ฅยณ โ 3๐ฅยฒ + 4๐ฅ + 6 is a polynomial in x. Similarly, ๐ฆยฒ + 2๐ฆ is a polynomial in the variable ๐ฆ and ๐ยฒ + 4 is a polynomial in the variable ๐.

Consider the expression, 3๐ฅ 2 + โ2๐ฅ This has square root of 2 in it. But the operations on the variable x involve only squaring and multiplication by fixed numbers 3 and โ2. So, this is a polynomial.

Consider another expression, 7 ๐ฅ+ ๐ฅ This involves the operation of taking reciprocal of the variable and so it is not a polynomial.

9

1.5 POLYNOMIALS โ RELATED TERMINOLOGIES

The different terms related to polynomials are given below:

Terms In an expression, a term can be a variable or constant or a product of variable and constant. In the polynomial ๐ฅยณ + 2๐ฅ, the expressions ๐ฅยณ and 2๐ฅ are called the terms of the polynomial Coefficient A coefficient is a numerical value, which is written along with a variable. So, in 8๐ฅ 3 โ 4๐ฅยฒ + 5๐ฅ โ 3, the coefficient of ๐ฅยณ is 8, the coefficient of๐ฅยฒ is โ4, the coefficient of ๐ฅ is 5 and โ3 is the coefficient of ๐ฅ 0 . Variable A variable is a letter that represents the unknown value in an expression. In ๐ฅยฒ + 2๐ฅ, the variable is ๐ฅ.

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Constant A constant is a number, whose value never changes in an expression. Consider an example, 5๐ฅ + 2 Here, The variable is ๐ฅ, coefficient is 5, constant is 2, and terms are 5๐ฅ and 2.

If the variable in a polynomial is ๐ฅ, we may denote the polynomial by ๐(๐ฅ), or ๐(๐ฅ), or ๐(๐ฅ), etc. So, for example, we may write: ๐(๐ฅ ) = ๐ฅ 3 + 5๐ฅ โ 3 ๐(๐ฅ) = 3๐ฅยฒ โ 1 ๐(๐ฆ) = ๐ฆยณ + 4๐ฆ + 2

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1.4 DEGREE OF A POLYNOMIAL What is degree of a polynomial? The degree of a polynomial is the highest power of the variable in a polynomial expression.

For instance, ๐(๐ฅ ) = 6๐ฅยณ + 2๐ฅยฒ + 4 is a polynomial. Here 6๐ฅยณ, 2๐ฅยฒ, 4 are the terms where 6๐ฅยณ is a leading term and 4 is a constant term. The coefficients of the polynomial are 6 and 2. The degree of the polynomial 6๐ฅยณ + 2๐ฅยฒ + 4 ๐๐ 3. Or, 6๐ฅยณ + 2๐ฅยฒ + 4 is a polynomial of degree 3. Similarly, for the polynomial ๐(๐ฅ ) = 3๐ฅ 8 + 4๐ฅ 3 + 9๐ฅ + 1 The degree is 8. The polynomial 3๐ฅ 8 + 4๐ฅ 3 + 9๐ฅ + 1 is a polynomial of degree 8.

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Rather than saying polynomial of degree 1, polynomial of degree 2, polynomial of degree 3 and so on, we can say first degree polynomial, second degree polynomial, third degree polynomial and so on. Based on the degree, we can write the general forms of all polynomials.

First degree polynomial Second degree polynomial Third degree polynomial

๐๐ฅ + ๐ ๐๐ฅยฒ + ๐๐ฅ + ๐ ๐๐ฅยณ + ๐๐ฅ + ๐๐ฅ + ๐

Here, the letters ๐, ๐, ๐, ๐ are coefficients.

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ACTIVITY 1. Write the coefficients of ๐ฅยฒ in each of the following: (i) 3๐ฅยฒ + ๐ฅ (ii) 12 โ ๐ฅยฒ + ๐ฅ (iii) 2๏ฐ ยณ + 2๐ฅยฒ (iv) โ2๐ฅ 2 + 3๐ฅ + 4

2. Which of the following are polynomials (i) 5x (ii) 0๐ฅยฒ + 2๐ฅ + 4 (iii) โ2๐ฅ + 3๐ฅ 1 2 1 (iv) ๐ฅ + 2 4๐ฅ 3.Write the degree of each of the following polynomials: (i) ๐ฅยณ + 8๐ฅ + 7๐ฅยฒ (ii) 1 โ ๐ฆยฒ (iii) 2๐ก โ 4 (iv) 9๐ฆ

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4.Find ๐(1) and ๐(2) in the following polynomials, (i) ๐(๐ฅ) = ๐ฅ + 5 (ii) ๐(๐ฅ) = 5๐ฅ 2 + 2 โ ๐ฅ (iii) ๐(๐ฅ) = 2๐ฅ 3 + 3๐ฅ 2 + 4๐ฅ + 7 5.Write each of the operations below in an algebraic form and state whether it is a polynomial or not. (i) (ii) (iii) (iv)

Sum of a number and its square. Sum of a number and its reciprocal. Product of the sum and difference of a number and its square root. Sum of a number and its square root.

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Jishnusagar V S S III B.Ed. Mathematics Karmela Rani Training College Kollam 1

NATIONAL ANTHEM Jana Gana Mana Adhinayaka Jay He Bharata Bhagya Vidhata Panjab Sindhu Gujarat Maratha Dravida Utkala Banga Vindhya Himachal Yamuna Ganga Uchchala Jaladhi Taranga Tava Subha Name Jaage Tave Subha Aashish Mange Gaahe Tava Jay Gaatha Jana Gana Mangal Daayak Jay He Bharat Bhagya Vidhata Jay he Jay he Jay he Jay Jay Jay Jay he

PLEDGE India is my country and all Indians are my brothers and sisters. I love my country and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give respect to my parents, teachers and elders and treat everyone with courtesy.to my country and my people, I pledge my devotion. In their wellbeing and prosperity alone, lies my happiness.

2

PREAMBLE OF INDIAN CONSTITUTION

3

Dear students, Starting your day with a positive attitude towards

learning and your life of study can be an enormous help to make your day in school go smoothly. Just a few minutes of reflection and writing down one positive thought in a gratitude journal every day will gradually form a more positive outlook. You might not see the effects immediately but if you tend to worry, try it and see the difference it can make to your day. May your journey through this book make learning a joyful experience. With love and regard, Jishnusagar V S

4

CHAPTER 1

POLYNOMIALS 1.1

Introduction

1.2

Polynomials in one variable

1.3

Polynomials โ Related Terminologies

1.4

Degree of a polynomial

5

POLYNOMIALS

1.1 INTRODUCTION

What is a polynomial The word polynomial is derived from the Greek words poly(meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms". Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.

6

1.2 POLYNOMIALS IN ONE VARIABLE Polynomials in one variable is an algebraic expression that consists of one variable in it. Some of the examples of polynomials in one variable are given below

โข

๐ฅ+2

โข

๐ฅยฒ + 3๐ฅ + 2

โข

๐ฆยณ + 2๐ฆยฒ โ ๐ฆ + 5

Recall that a variable is denoted by a symbol that can take any real value. We use the letters ๐ฅ, ๐ฆ, ๐ง, etc. to denote variables. 2๐ฅ, โ ๐ฅ, โ 1/2 ๐ฅ are all algebraic expressions These expressions are of the form (a constant) ร ๐ฅ. Now suppose we want to write an expression which is (a constant) ร (a variable) and we do not know what the constant is. In such cases, we write the constant as ๐, ๐, ๐, etc. So, the expression will be ๐๐ฅ, say.

7

Fig 1.1 2

2 Consider a square of side 2 units (see Fig. 1.1). What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its four sides. Here, each side is 2 units. So, its perimeter is 4 ร 2, i.e., 8 units. What will be the perimeter if each side of the square is 10 units? The perimeter is 4 ร 10, i.e., 40 units. In case the length of each side is x units (see Fig. 2.2), the perimeter is given by 4๐ฅ units. So, as the length of the side varies, the perimeter varies. Fig 1.2

๐ฅ 8

If you find the area of the square PQRS? It is ๐ฅ ร ๐ฅ = ๐ฅ 2 square units. = ๐ฅ 2 is an algebraic expression. You are also familiar with other algebraic expressions like 2๐ฅ, ๐ฅยฒ + 2๐ฅ, ๐ฅยณโ ๐ฅยฒ + 4๐ฅ + 7. Note that, all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable. Expressions of this form are called polynomials in one variable. For instance, 2๐ฅยณ โ 3๐ฅยฒ + 4๐ฅ + 6 is a polynomial in x. Similarly, ๐ฆยฒ + 2๐ฆ is a polynomial in the variable ๐ฆ and ๐ยฒ + 4 is a polynomial in the variable ๐.

Consider the expression, 3๐ฅ 2 + โ2๐ฅ This has square root of 2 in it. But the operations on the variable x involve only squaring and multiplication by fixed numbers 3 and โ2. So, this is a polynomial.

Consider another expression, 7 ๐ฅ+ ๐ฅ This involves the operation of taking reciprocal of the variable and so it is not a polynomial.

9

1.5 POLYNOMIALS โ RELATED TERMINOLOGIES

The different terms related to polynomials are given below:

Terms In an expression, a term can be a variable or constant or a product of variable and constant. In the polynomial ๐ฅยณ + 2๐ฅ, the expressions ๐ฅยณ and 2๐ฅ are called the terms of the polynomial Coefficient A coefficient is a numerical value, which is written along with a variable. So, in 8๐ฅ 3 โ 4๐ฅยฒ + 5๐ฅ โ 3, the coefficient of ๐ฅยณ is 8, the coefficient of๐ฅยฒ is โ4, the coefficient of ๐ฅ is 5 and โ3 is the coefficient of ๐ฅ 0 . Variable A variable is a letter that represents the unknown value in an expression. In ๐ฅยฒ + 2๐ฅ, the variable is ๐ฅ.

10

Constant A constant is a number, whose value never changes in an expression. Consider an example, 5๐ฅ + 2 Here, The variable is ๐ฅ, coefficient is 5, constant is 2, and terms are 5๐ฅ and 2.

If the variable in a polynomial is ๐ฅ, we may denote the polynomial by ๐(๐ฅ), or ๐(๐ฅ), or ๐(๐ฅ), etc. So, for example, we may write: ๐(๐ฅ ) = ๐ฅ 3 + 5๐ฅ โ 3 ๐(๐ฅ) = 3๐ฅยฒ โ 1 ๐(๐ฆ) = ๐ฆยณ + 4๐ฆ + 2

11

1.4 DEGREE OF A POLYNOMIAL What is degree of a polynomial? The degree of a polynomial is the highest power of the variable in a polynomial expression.

For instance, ๐(๐ฅ ) = 6๐ฅยณ + 2๐ฅยฒ + 4 is a polynomial. Here 6๐ฅยณ, 2๐ฅยฒ, 4 are the terms where 6๐ฅยณ is a leading term and 4 is a constant term. The coefficients of the polynomial are 6 and 2. The degree of the polynomial 6๐ฅยณ + 2๐ฅยฒ + 4 ๐๐ 3. Or, 6๐ฅยณ + 2๐ฅยฒ + 4 is a polynomial of degree 3. Similarly, for the polynomial ๐(๐ฅ ) = 3๐ฅ 8 + 4๐ฅ 3 + 9๐ฅ + 1 The degree is 8. The polynomial 3๐ฅ 8 + 4๐ฅ 3 + 9๐ฅ + 1 is a polynomial of degree 8.

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Rather than saying polynomial of degree 1, polynomial of degree 2, polynomial of degree 3 and so on, we can say first degree polynomial, second degree polynomial, third degree polynomial and so on. Based on the degree, we can write the general forms of all polynomials.

First degree polynomial Second degree polynomial Third degree polynomial

๐๐ฅ + ๐ ๐๐ฅยฒ + ๐๐ฅ + ๐ ๐๐ฅยณ + ๐๐ฅ + ๐๐ฅ + ๐

Here, the letters ๐, ๐, ๐, ๐ are coefficients.

13

ACTIVITY 1. Write the coefficients of ๐ฅยฒ in each of the following: (i) 3๐ฅยฒ + ๐ฅ (ii) 12 โ ๐ฅยฒ + ๐ฅ (iii) 2๏ฐ ยณ + 2๐ฅยฒ (iv) โ2๐ฅ 2 + 3๐ฅ + 4

2. Which of the following are polynomials (i) 5x (ii) 0๐ฅยฒ + 2๐ฅ + 4 (iii) โ2๐ฅ + 3๐ฅ 1 2 1 (iv) ๐ฅ + 2 4๐ฅ 3.Write the degree of each of the following polynomials: (i) ๐ฅยณ + 8๐ฅ + 7๐ฅยฒ (ii) 1 โ ๐ฆยฒ (iii) 2๐ก โ 4 (iv) 9๐ฆ

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4.Find ๐(1) and ๐(2) in the following polynomials, (i) ๐(๐ฅ) = ๐ฅ + 5 (ii) ๐(๐ฅ) = 5๐ฅ 2 + 2 โ ๐ฅ (iii) ๐(๐ฅ) = 2๐ฅ 3 + 3๐ฅ 2 + 4๐ฅ + 7 5.Write each of the operations below in an algebraic form and state whether it is a polynomial or not. (i) (ii) (iii) (iv)

Sum of a number and its square. Sum of a number and its reciprocal. Product of the sum and difference of a number and its square root. Sum of a number and its square root.

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