NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials

NCERT Solutions for Class 9 Maths Chapter 2, "Polynomials", consists of three exercises, each covering a specific set of questions. Below is a detailed explanation of each exercise:

• Exercise 2.1: This exercise covers the definition and basic concepts of polynomials. The questions in this exercise aim to familiarise students with terms like coefficients, exponents, degrees, and standard forms of polynomials. Students are also required to classify polynomials based on their degrees. They will have to solve problems related to the addition, subtraction, and multiplication of polynomials and also learn how to factorise polynomials.

• Exercise 2.2: This exercise deals with the factors and zeros of polynomials. The questions in this exercise require students to find the factors and zeros of given polynomials. They will also learn how to use the factor theorem and remainder theorem to factorise polynomials and find their zeros.

• Exercise 2.3: This exercise covers the division algorithm for polynomials. The questions in this exercise require students to divide a polynomial by another polynomial using the long division method. They will also learn how to use the remainder theorem to find the remainder when a polynomial is divided by another polynomial.

NCERT Solutions for Class 9 Maths Chapter 2 Polynomial - PDF Download

You can opt for Chapter 2 - Polynomials NCERT Solutions for Class 9 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 9 Maths

• Chapter 1 - Number System

• Chapter 3 - Coordinate Geometry

• Chapter 4 - Linear Equations in Two Variables

• Chapter 5 - Introductions to Euclids Geometry

• Chapter 6 - Lines and Angles

• Chapter 7 - Triangles

• Chapter 8 - Quadrilaterals

• Chapter 9 - Areas of Parallelogram and Triangles

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Herons formula

• Chapter 13 - Surface area and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Polynomials

Chapter 2 Polynomials Class 9 is divided into six sections and five exercises. The first section is the introduction with no exercise. The second section discusses a particular type of algebraic expression called polynomials. The third section explains the zeros of polynomials whereas the Fourth section discusses the Remainder Theorem. Factorization of Polynomials and identities are revisited in the fifth and sixth sections.

NCERT Solutions Class 9 Maths Chapter 2 Exercises

Variables - The Unknown Value

Have you ever wondered why children have different heights? Some children grow taller and some end up being shorter than average. To answer this question Scientists have come closer and researched the parameters in the form of variables that are the cause of height. 

The word ‘variable’ is derived from the word ‘vary’ which means changing. Therefore, a variable can be any trait, condition, or factor that can change by only differing amounts or it is the unknown term whose value is not known. Example: A child’s height is dependent on the amount of protein and nutrients he or she consumes. Not only that, the height of kids is also dependent on their DNA which means if their parents are tall then there are more chances of them being tall whereas short parents usually have short kids. The height of the kid is also dependent on the rate of work or activities. It is believed that children with more activities like jumping, running, skipping, etc tend to grow faster. Thus, nutrients, DNA, and activities are the three variables that control the height in our body. These variables keep changing from body to body. 

For example, while cooking dal we know that the quantity of water is thrice the number of lentils. That you can add 1 cup of lentils to three cups of water. This process can be expressed as,

“3x + x”  

Here, the quantity of lentils is variable. That means if the quantity of lentils changes then the quantity of water also changes. 

In a World Full of Variables, You Will Always Find Constant.

There is one interesting thing about constants and that is this it never changes. A constant is actually a value that is a fixed number on its own. For example - In the equation 9 - x = 5, 9 and 5 are two constants whose values will not change whereas the value of x is not known. Thus, x is a variable.

Can Constant be a Coefficient Too?

Since now we already know about variables, it is easier for us to understand the constant and coefficient. A coefficient is usually the number that is multiplied by the variable or letters. For example in ‘5x + y - 7’, 5 is a coefficient of x in the term 5x because it is a number that is multiplied by the unknown variable x. Also, in the term y, it can be considered as the coefficient of y because y can be written as 1xy. 

The coefficient is the number that is always multiplied by the variables but constants are terms without variables. Therefore, coefficients cannot be called constants and vice versa. In the aforementioned example, -7 is constant.

Like Terms

Like terms are the terms having the same variables raised to the same power. In 5x + y - 7, no variable is common therefore no like terms. 

In 5a + 2b - 3a + 4 the terms like 5a and -3a are like terms whereas 4 is constant.

What is a Polynomial?

The word Polynomial is derived from the word poly ("many") and nominal ("term"). It is an expression consisting of many terms such that each term holds at least one variable. The variables can be raised to the power and further multiplied by a coefficient but the simplest polynomials hold one variable. The terms are separated by signs( + or - ). Also, the variables and numbers can be combined using addition, subtraction, multiplication or division but it can never be divided by a variable which means a term can never be like  2/x. A polynomial can also not have infinite terms. It always has a finite sum of terms with all variables having whole-number exponents and no variable as a denominator. 

Polynomials are composed of the following:

• Constants such as 3, −20, or ½, etc.

• Variables such as g, h, x, y, etc.

• Exponents such as 2 in y2 or 5 in x5  etc

Examples of Polynomials: 5x3 – 2x2 + x – 13 and  x2y3 + xy.

Degree of a Polynomial

It is simply the highest of the powers or exponents on the terms present in the algebraic expression.

Example: In 7x – 5, the first term is 7x, whereas the second term is -5. The power on the variable of the given first term is one and on the second term is zero. Since the highest exponent is one, the degree of the polynomial is also 1.

Types of Polynomials

Polynomials can be classified on the basis of

1. Number of terms.

2. Degree of a polynomial.

Classification on the Basis of Terms

A polynomial either has one term, two terms, three terms, or more than three terms.

1. Monomials- ‘Mono’ stands for one and ‘mial’ stands for terms thus an algebraic expression with one term is called a monomial. 

2. Binomials- ‘Bi’ stands for two and ‘mial’ stands for terms therefore an algebraic expression with two, unlike terms is called binomials. 

3. Trinomials- ‘Tri’ stands for three and ‘mial’ stands for terms thus an algebraic expression with three unlike terms is called trinomials. 

Classification on the Basis of Degrees

The Degree of Polynomial is considered as the highest value of the exponent in the expression because it is the largest exponent. We can also call it an order of the polynomial. While finding the degree of the polynomial, remember that the polynomial powers of the variables must be either in an ascending or descending order.

1. Linear Polynomial: If the expression holds degree 1 then we can call it a linear polynomial. 

2. Quadratic Polynomial: If the expression holds degree 2 then it can be called a quadratic polynomial.

3. Cubic Polynomial: If the expression holds degree 3 then it will be called a cubic polynomial.

Zeros of Polynomials

If the value of every coefficient of a variable is zero then it is called the zeros of a Polynomial. In order to find the relationship between the zeroes and coefficients of a given quadratic polynomial, we can find the zeros of the polynomial by the factorization method that is, by taking the sum and product of these zeros.

Operations on Polynomial

There are four main polynomial operations which are:

• Addition of Polynomials

• Subtraction of Polynomials

• Multiplication of Polynomials

• Division of Polynomials

Every operation on polynomial class 9 is explained below using solved examples.

Practice To Win - NCERT Solutions 

To pass the CBSE test, tough preparation and devotion are required, which may be accomplished with the correct amount of hard effort and persistent practise. It is also critical to work carefully and plan your preparedness. After studying the chapter, doing math questions from CBSE Class 9 Mathematics Chapter 2 polynomials solutions helps you grasp the test format and weightage of each topic. Remember that it is a practise that will improve your speed during tests. Here is where many students fail to finish their papers on time. Practice from Ch 2 Mathematics Class 9 instills in you the proclivity to solve the provided question at breakneck speed.

There is always a time crunch while we prepare for exams. NCERT Solutions for Class 9 Maths Chapter 2 Polynomials serve as study material to help in the fast and systematic revision of the chapter while preparing for the exam. It is less time-consuming and covers the entire syllabus. Practicing from NCERT Solutions for Class 9 Maths Chapter 2 pdf helps you to rectify your mistakes and ease your struggle to solve the chapter because it is very important to fast-track your preparation. 

1. Revision in the Form of Question-Answer

It serves as a revision note as you get the opportunity to revise all the chapters you have studied in a question-answer pattern which is indeed more beneficial than you reading the notes. Revision in the question-answer pattern will help you grab or recollect the important points that you might have missed while reading the chapter. In other words, it brushes and summarizes the chapter to make it easier for you to remember. Therefore, refer to the NCERT Class 9 Maths Chapter 2 Solutions and try to improve the areas where you are weak by rectifying your mistakes.

2. Analyzation and Strategizing the Preparation.

It helps you to understand and it mirrors how strong your preparation is by testing your knowledge. Often  we study the whole chapter very well but we still fail to answer the questions. This can be because of various reasons like you forgetting due to lack of practice and revision or you missing out on the important points while superficially reading the chapter, etc. Solving question papers and answering random questions from different chapters can bring you to the realization of how strong your preparation is. This will help you to strategize your preparation in order to extract maximum marks from what you have already studied.

3. Interpretation of Important Questions and the Complexity Level.

Even if CBSE changes the question paper pattern in exam 2024, it is important for you to become cognizant of the fact that the concept of the chapter and the significance of the topics remains the same. Solving previous year's question papers will help you know the topics holding significant weightage in a chapter. Moreover, you can examine the magnitude of the complexity of the questions asked during board exams to prepare yourself accordingly.

4. Improvisation of Speed through Practice. 

Practice makes you confident in what you study and delivers perfection in your answer. It is always suggested to practice the important topics and questions again and again so that you don’t forget during the exam. Solving NCERT Class 9 Maths Chapter 2 involves answering the same questions in different ways that eventually leads to memorization of the concept and thus increases your speed.

5. Guess the Important Chapters.

The paper you solve the more you get hold of the technique used in preparing the question paper. You get to notice the number of questions that are repeated and the interval of repetition. On Gauging NCERT Solutions Chapter 2 Class 9 Maths you become aware of the topics that haven’t come in the exam for a long time and therefore have a high chance of coming in the next exam. This way you can guess the important topics, even if the exact same question doesn’t come but the concept remains the same. 

NCERT Solutions By Vedantu - A Helping Hand In Your Preparation.

Vedantu is an ardent believer in smart work and harbors experienced teaching professionals who are adept at learning and possess a greater passion for imparting the same. Vedantu makes the learning experience fun by offering solutions in a step-by-step explanation of numerical problems to help you improve your understanding of the concept related to the topics. This solution is engineered by the experts of Vedantu to serve it as an excellent material for practice and make the learning process more convenient. 

Vedantu promotes the smartest way of learning by facilitating the NCERT Solutions for Class 9 Maths with the following benefits:

1. Properly standardized solutions It contains comprehensive answers to all the questions

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3. Carefully crafted by professionals. 

4. Help you hold a strong grasp on the subject

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6. Prepared and reviewed by experienced subject matter experts.

7. It is revised according to the latest NCERT syllabus and guidelines of the Central Board of Secondary Education (CBSE).

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9. Serves as a good Revision Note 

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11. It is written keeping in mind the age group of the students.

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14. It gives a gist of the entire chapter and concept in the form of solutions.

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NCERT Solutions for Class 9 Maths Chapter 2 Polynomials - Free PDF

Vedantu tried its best to render you real help by providing the NCERT Solutions for Class 9th Maths Chapter 2 Polynomial. It aimed to deliver sufficient problems and solutions to practice and build a strong foundation for the chapter. 

Deeper Into The Exercises

Polynomials class 9 have a total of four exercises however the last exercise is optional. In the first exercise, you need to find zeros of polynomials p(x). In the second exercise, there are two questions. You have to verify the relationship between the zeroes and the coefficients in the first, while in the second, you have to find a quadratic polynomial. The third exercise has a total of five questions in which you have to do the division of polynomials and obtain zeros of polynomials. Then comes the optional exercise which has five questions in which you need to find zeros of polynomials.

Vedantu provides all NCERT Answers to these exercises in PDF format for Chapter 2 Class 9 Mathematics. Each lesson is followed by a short set of activities. The activities are designed to assess your knowledge and depth of comprehension of the many theorems and concepts discussed in this chapter. However, the numerical problems in this chapter are generally dependent on particular theorems and other related topics.

To further help you improve your understanding of these topics and related concepts, numbers of solved examples of numerical problems are also offered. Moreover, a thorough step-by-step explanation is provided for each solved example. It can help understand which methods are to be used to approach different types of questions for solving them accurately. 

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials - PDF Download

The Vedantu team has verified the exercise questions and types of questions in class 9th Maths Chapter 2. NCERT Solutions for Class 9 Maths Chapter 2 free pdf of Polynomials solved by experts are available on Vedantu. Download Free Study Material of Ch 2 Maths Class 9 to score more marks.

What Are The Benefits of Having Class 9 Maths Chapter 2 Solutions Before Exams?

Class 9 Chapter 2 Maths Solutions available with Vedantu are to provide a basic understanding and simple and easy solutions. Students feel relaxed after having clarification to each of the solving steps and proper explanation. For CBSE students, NCERT Class 9 Maths Chapter 2  is in demand all the time. The PDF available with Vedantu has numerous solved questions of different types so that students might not feel exhausted while giving a final attempt in the exam. 

NCERT solutions Class 9 Maths Chapter 2 is prepared in such a way that it covers the entire syllabus with a proper step-by-step explanation. Also, the primary focus is on developing troubleshooting and problem-solving skills among students. Attempting an exam is a faraway concept; the basics are all about learning and understanding. It is important to avoid the last-minute rush to prepare for the mathematics exam. Thus Chapter 2 Class 9 Maths solutions will surely help them to score good marks. 

The Different Benefits of Referring to Polynomials Class 9 Ncert Solutions Include

• You can find a solution to every kind of problem you face with a proper explanation.

• There are a total of six sections and five exercises in the polynomial chapter of Class 9. However, the basic concept of solution revolves around unknown variables, calculation, and finding a solution to unknown variables. But the techniques might differ in all aspects. 

• Practice is the key aspect to achieve good results. However, for that, you need proper solutions to check your steps. These NCERT Maths Class 9 Chapter 2 Solutions can help you with the best results and outcomes.