class 10 maths ch 2 assignment

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

Class 10 Maths Exercise 2.1 Solutions Class 10 Maths Exercise 2.2 Solutions

Get the NCERT Solutions for class 10 Maths chapter 2 Polynomials all exercises in Hindi and English medium modified and updated for session 2024-25. According to new syllabus and latest NCERT textbooks issued for academic year 2024-25, there are only two exercises in chapter 2 in class 10th mathematics syllabus.

Class 10 Maths Chapter 2 Solutions for CBSE Board

Class 10 Maths Chapter 2 Exercise 2.1
Class 10 Maths Chapter 2 Exercise 2.2

Class 10 Maths Chapter 2 Solutions for State Boards

Class 10 Maths Chapter 2 Exercise 2.3
Class 10 Maths Chapter 2 Exercise 2.4
Class 10th Maths Chapter 2 NCERT Book
Class 10 Mathematics Solutions
Class 10 all Subjects NCERT Solutions

Class 10 Maths Chapter 2 Polynomials, covers various important concepts related to polynomial equations. Here are the main points to learn from this chapter. Understand what polynomials are and their characteristics. A polynomial is an algebraic expression consisting of variables, coefficients, and exponents, with non-negative integer exponents. Learn how to determine the degree of a polynomial based on the highest power of the variable term. Identify polynomials as constant, linear, quadratic, cubic, etc., based on their degree.

The textbooks issued by NCERT shows that there are only two exercises in chapter 2 of 10th mathematics. So, students have to practice only exercise 2.1 and 2.2 for the board exams. According to new syllabus and books issued for new session, the course structure of class 10 chapter 2 is as follows: Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials.

1. Now, Number of Exercises: 2 2. Number of Periods needed: 8 3. Weightage of Maths: 4 – 5

Know how to add and subtract polynomials by combining like terms. Understand the process of multiplying polynomials using the distributive property. Learn how to use the FOIL method for binomial multiplication (First, Outer, Inner, Last). Multiply a monomial by a polynomial and apply the distributive property. Recognize and apply formulas for special products such as the square of a binomial (a + b)² and the difference of squares (a² – b²).

UP board Students, who are using NCERT Textbooks for their final exams, they can download UP Board Solutions for Class 10 Maths Chapter 2 from this page in PDF format. Assignments and Revision tests are given below in the format of PDF file.

Study polynomial division using long division and synthetic division methods. Understand the concepts of divisors, dividends, quotients, and remainders. Learn techniques for factoring polynomials, including common factors, grouping, and the use of algebraic identities. Factorize quadratic trinomials of the form ax² + bx + c.

Familiarize yourself with important algebraic identities, such as (a + b)², (a – b)², and a² – b². Understand the Remainder Theorem, which relates the remainder of polynomial division to the value of the polynomial at a given point. Know the Factor Theorem, which states that (x – a) is a factor of a polynomial if and only if the polynomial evaluates to zero at x = a.

Learn how to use synthetic division for polynomial division, particularly for dividing by linear divisors of the form (x – a). Apply polynomial equations to real-world problems and mathematical modeling. Solve polynomial equations, both linear and quadratic, by factoring and applying the zero-factor property. Understand the graphical representation of polynomial functions and how to find the roots or solutions by analyzing the graph.

Class 10 Maths Chapter 2 Practice Tests with Answers

Class 10 Maths Chapter 2 Practice Test 1
Class 10 Maths Chapter 2 Practice Test 2
Class 10 Maths Chapter 2 Practice Test 3
Class 10 Maths Chapter 2 Practice Test 4
Class 10 Maths Chapter 2 Practice Test 5
Class 10 Maths Chapter 2 Practice Test 6

According to real-time information, and the knowledge based on information available up to session 2024-25, we provide specific information about the availability of study materials for Class 10 Maths Chapter 2 Polynomials at Tiwari Academy website and Apps. Summarize the key concepts, formulas, and techniques learned in this chapter for quick reference. These are the main points to learn from Class 10 Maths Chapter 2 Polynomials. It is a fundamental chapter in algebra that serves as a basis for more advanced topics in mathematics.

What are the important terms of 10th Maths Chapter 2 Polynomials?

What do you understand by a polynomial as per class 10 maths chapter 2.

An algebraic expression, in which variable(s) does (do) not occur in the denominator, exponents of variable(s) are whole numbers and numerical coefficients of various terms are real numbers, is called a polynomial.

In other words,

1. No term of a polynomial has a variable in the denominator; 2. In each term of a polynomial, the exponents of the variable(s) are non-negative integers and 3. Numerical coefficient of each term is a real number.

The terms of a polynomial, having the same variable(s) and the same exponents of the variable(s), are called like terms. A polynomial of degree 2 is called a quadratic polynomial. The degree of a non-zero constant polynomial is taken as zero. When all the coefficients of variable(s) in the terms of a polynomial are zeros, the polynomial is called a zero polynomial. The degree of a zero polynomial is not defined.

Do you know what is a monomial?

An algebraic expression or a polynomial, consisting of only one term, is called a monomial.

Which type of polynomial is called a binomial?

An algebraic expression or a polynomial, consisting of only two terms, is called a binomial.

Which type of algebraic expression is known as trinomial?

An algebraic expression or a polynomial, consisting of only three terms, is called a trinomial.

Tiwari Academy and similar educational websites often provide NCERT solutions and study materials to assist students in their preparation. The reasons for using Tiwari Academy platforms for NCERT solutions includes in simplified format. This platform offers comprehensive solutions that cover all the exercises and examples from the NCERT textbook, making it easier for students to understand and practice.

Many platforms provide step-by-step explanations for solving problems, helping students grasp the concepts more effectively. But they provide best and confined to current syllabus. In addition to NCERT solutions, Tiwari Academy platforms often offer additional practice questions and exercises to help students gain confidence and excel in their studies.

Online resources like Tiwari Academy are easily accessible, allowing students to study at their own pace and convenience. These platforms may also provide supplementary materials such as video tutorials, quizzes, and sample papers to enhance learning. Tiwari Academy and similar platforms may offer study materials and solutions for a wide range of subjects and topics, making them a one-stop destination for students’ academic needs.

Class 10 Exercises solutions are solved in both English as well as Hindi medium in order to help all type of students following latest CBSE Syllabus 2024-25. In prashnavali 2.1 and 2.2 Ganit Solutions, if there is any inconvenient to understand, please inform us, we will short out at our level best.

If you are considering using Tiwari Academy or a similar platform for Class 10 Maths Chapter 2 Polynomials, I recommend visiting their website and exploring the available resources to see and get it as per your learning needs and preferences. Additionally, you can seek recommendations from teachers or peers to determine the quality and effectiveness of the materials provided by such platforms.

The value(s) of the variable for which the value of a polynomial in one variable is zero is (are) called zero(s) of the polynomial. To verify the relationship between the zeroes and coefficients of a given quadratic polynomial, we can find the zeroes of p(x) by factorisation. By taking sum and product of these zeros, we can verify the following results.

Historical Facts about Polynomials

An elegant way of dividing a polynomial by a liner polynomial was introduced by Paolo Ruffin in 1809. His method is known as Synthetic division, which facilitates the division of a polynomial by a linear polynomial or binomial of the form x – a with the help of the coefficients involved.

Determining the zeros of polynomials, or finding roots of algebraic equations is among the oldest problems in mathematics. The modern way, we use today only developed beginning in the 15th century. Before that, linear equations were written out in words.

1. The use of the equal to (=) sign is in Robert Recorde’s book (The Whetstone of Witte in 1557). Plus sign (+) the sign of addition, minus sign (−) the sign of subtraction and the use of an alphabet for an unknown variable in Michael Stifel’s book (Arithemetica integra in 1544).

2. René Descartes, in 1637, introduced the concept of plotting the graph of a polynomial equation. Just because of him, the popularity of use of letters of the alphabet to denote constants and letters from the end of the alphabet (x, y, z, etc.) to denote variables (like 2x, 3y, 7z, etc.) in the general formula for a polynomial in one variable.

How to make good Practice in Class 10 Maths Chapter 2 Polynomials?

Polynomials are said to be one of the important sections of mathematics, especially for Algebra. They’re used in most of the equations and topics of numerical operation in standard 10 and higher also. It wouldn’t be wrong, if we say polynomials are Building Blocks of Maths expressions. Here we will learn about how to solve all exercises of 10th Maths chapter 2. Before learning about the polynomial equation, we have to know its significant importance that makes the topic more interesting. Many Maths process that are done in everyday life can be interpreted as polynomials. With the help of polynomial equations, one can calculate the grocery bill for small and even distance travelled by light in space.

Step 1: NCERT Solutions for Class 10 Maths Chapter 1 with basic knowledge of Polynomial.

Step 2: class 10 maths chapter 2 solutions with relationship between zeros and coefficient of variables., step 3: ncert class 10 maths chapter 2 solutions to prove an irrational number as irrational., step 4: class x maths chapter 2 solution help to identify the terminating and non-terminating decimals., step 5: class 10 maths chapter 2 need to build structural approach towards learning..

What are real life applications of class 10th mathematics chapter 2?

Some real life applications of class 10th mathematics chapter 2 (Polynomials) are:

Polynomials can be used to model different types of situations, like in the stock market to see how prices will vary with time.
In physics also polynomials are used to describe the trajectory of projectiles.
Polynomials used in industries and construction field also. Polynomials are useful for every person and in every field.

How many exercises are there in chapter 2 of 10th Maths?

There are in all 2 exercises in class 10 mathematics chapter 2 (Polynomials). In first exercise (Ex 2.1), there is only 1 question having 6 parts. In second exercise (Ex 2.2), there are 2 questions and each question have 6 parts. So, there are total 3 questions in class 10 mathematics chapter 2 (Polynomials). In this chapter there are in all 9 examples. Example 1 is based on Ex 2.1, Examples 2, 3, 4, 5 are based on Ex 2.2.

Which questions and examples are important in Class 10 Maths Chapter 2?

In first exercise (Ex 2.1) there is only 1 question with 6 parts and all the part of this question are equally important. In second exercise (Ex 2.2) all questions are important. Important examples of chapter 2 (Polynomials) class 10th mathematics are example 1, 2, 3, 4, 8, 9.

What should we recall before starting chapter 2 of 10th Maths?

Before starting class 10th mathematics chapter 2 (Polynomials), students should recall chapter 2 (Polynomials) of class 9th mathematics.

« Chapter 1: Real Numbers

Chapter 3: pair of linear equations in two variables ».

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Chapter 2 Class 10 Polynomials

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Updated the chapter according to the new NCERT Book for 2023-24 session

Get NCERT Solutions of C hapter 2 Class 10 Polynomials free at Teachoo. All NCERT Exercise Questions, Examples and Optional Questions have been solved with video of each and every question.

In this chapter, we will learn

What is a polynomial
What are monomial, binomials, trinomials
What is the degree of polynomial
What are linear, quadratic and cubic polynomials
What are Zeroes of a Polynomials

How to find Zeroes of a Polynomials (both quadratic and cubic)
Quadratic Polynomial
Cubic Polynomial
Dividing two polynomials, and verifying the Division Algorithm for Polynomials

We have divided this chapter into 2 parts - Serial Order Wise and Concept Wise.

In Serial Order Wise, there are examples and exercises just like the NCERT Book. It is useful when we are looking to get an answer of a particular question.

That is not a good way of doing the chapter.

The best way to learning maths is through Concept Wise

Concept wise is Teachoo's (टीचू) way of learning.

In concept wise, we have divided the chapter into concepts. First the concept is explained, and then the questions of that concept is solved - from easy to difficult.

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NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

Cbse class 10 mathematics ncert solutions: chapter 2, polynomials.

To get an overall learning of the topics studied in CBSE Class 10 Maths chapter- Polynomials, all you need is to practice the NCERT questions and check for their correctness by referring the NCERT solutions provided here. Class 10 Maths NCERT Solutions

Preparing for an exam requires an overall understanding of all the concepts and topics to be tested in that exam. This can be achieved by a thorough learning of each topic and an intensive practice of the questions based on those topics. When we talk about the board exams or other entrance exams, NCERT books can be considered as the best source for preparation because these books help to clear the concepts and offer a number of problems to assess your learning.

Get here the precise NCERT Solutions for Class 10 Maths

For all the questions given in CBSE Class 10 Mathematics NCERT book we have collated detailed and accurate answers that will help students learn the right technique to write perfect answers in the annual exams to get optimum marks. Moreover, these answers have been framed in step-wise mentioning each concept and formula implied in the solutions. This will make it easier for students to understand the technique implied to solve each NCERT question.

In this chapter we are providing the NCERT solutions for Class 10 Maths Chapter 2, Polynomials. All the solutions have been collated as PDF format which students may easily download without paying any charge.

Main topics discussed in Class 10 Mathematics chapter- Polynomials are:

Basics concepts related to polynomials
Geometrical meaning of the zeroes of a polynomial
Relationship between zeroes and coefficients of a polynomial
Division algorithm for polynomials and

Some of the questions and their solutions from NCERT Solutions for Class 10: Polynomials, are as follows:

To get the complete solution click on the following link:

CBSE Class 10 Maths Exam : Check Chapter-wise Weightage Paper with Latest Marking Scheme

Check here the chapter-wise weightage distribution for the CBSE Class 10 Maths Exam . Mark the chapters with high weightage to prepare them carefully. CBSE Class 10 Maths Chapter wise Weightage for Board Exam

CBSE Class 10 Maths Paper under the ongoing board exams will be conducted tomorrow – 12th March, . This year board will conduct two papers – Mathematics Standard and Mathematics Basic. On one side, where the students with basic Maths will solve the easy paper, the paper will be a bit tough for the students with standard Maths. In fact, the level of the standard Maths paper will be the same as of previous years’ board examinations. So, here the students with standard Maths will be more anxious about their preparations as they have to score good marks in the paper so that they may continue with Mathematics in the higher classes as well.

In this article, we are providing the in-depth analysis of the latest CBSE sample paper of Class 10 Standard Maths. With this analysis we are providing here the chapter-wise weightage distribution for the upcoming Maths Exam. This will help students to know the chapters which are important for the exam and need to be prepared carefully.

In CBSE Class 10 Maths Paper , there will be 40 questions divided into four sections – A, B, C and D.

Section A (1-20) – Objective type questions carrying 1 mark each
Section B (21-26) – Short answer type questions (Type I) carrying 2 marks each
Section A (27-34) – Short answer type questions (Type II) carrying 3 marks each
Section A (35-40) – Long answer type questions carrying 4 marks each

You can check below the number of questions from each chapter asked in all the four sections of Standard Maths Sample Paper:

According to the weightage mentioned above, we can now pick out the chapters which carry high weightage for the annual board examination.

Important Chapters with High Weightage for CBSE Class 10 Maths Board Exam

Chapter 3 – Pair of Linear Equations in Two Variables = 8 Marks
Chapter 5 – Arithmetic Progression = 5 Marks
Chapter 6 – Triangles = 9 Marks
Chapter 7 – Coordinate Geometry = 6 Marks
Chapter 8 – Introduction to Trigonometry = 6 Marks
Chapter 9 – Some Applications of Trigonometry = 6 Marks
Chapter 13 – Surface Areas and Volumes = 7 Marks
Chapter 14 – Statistics = 8 Marks

Thus students must focus on these important chapters to prepare them thoroughly so that they may secure high marks in their Maths paper.

Important thing to note here is that this analysis is completely based on the CBSE Class 10 Maths Sample Paper. Board, generally follows the same pattern as of sample papers, in the questions papers of board examinations. So, we can expect the questions in the Maths Paper arranged according to the distribution discussed here.

NCERT Solutions for class 10 Maths Chapter 2- Polynomials

As this is one of the important topics in maths, it comes under the unit – Algebra which has a weightage of 20 marks in the class 10 maths board exams. The average number of questions asked from this chapter is usually 1. This chapter talks about the following,

Introduction to Polynomials
Geometrical Meaning of the Zeros of Polynomial
Relationship between Zeros and Coefficients of a Polynomial
Division Algorithm for Polynomials

Polynomials are introduced in class 9 where we discussed polynomials in one variable and their degrees in the previous class and this is discussed more in details in class 10. The NCERT solutions for class 10 maths for this chapter discusses the answers for various types of questions related to polynomials and their applications. We study about the division algorithm for polynomials of integers and also whether the zeroes of quadratic polynomials are related to their coefficients.

The chapter starts with the introduction of polynomials in section 2.1 followed by two very important topics in section 2.2 and 2.3

Geometrical Meaning of the zeroes of a Polynomial – It includes 1 question having 6 different cases.
Relationship between Zeroes and Coefficients of a polynomial – Explore the relationship between zeroes and coefficients of a quadratic polynomial through solutions to 2 problems in Exercise 2.2 having 6 parts in each question.

Next, it discusses the following topics which were introduced in class 9.

Division Algorithm for Polynomials – In this, the solutions for 5 problems in Exercise 2.3 is given having three long questions.

CBSE NCERT Solutions

NCERT and CBSE Solutions for free

Class 10 Mathematics Assignments

We have provided below free printable Class 10 Mathematics Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 10 Mathematics . These Assignments for Grade 10 Mathematics cover all important topics which can come in your standard 10 tests and examinations. Free printable Assignments for CBSE Class 10 Mathematics , school and class assignments, and practice test papers have been designed by our highly experienced class 10 faculty. You can free download CBSE NCERT printable Assignments for Mathematics Class 10 with solutions and answers. All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Class 10. Students can click on the links below and download all Pdf Assignments for Mathematics class 10 for free. All latest Kendriya Vidyalaya Class 10 Mathematics Assignments with Answers and test papers are given below.

Mathematics Class 10 Assignments Pdf Download

We have provided below the biggest collection of free CBSE NCERT KVS Assignments for Class 10 Mathematics . Students and teachers can download and save all free Mathematics assignments in Pdf for grade 10th. Our expert faculty have covered Class 10 important questions and answers for Mathematics as per the latest syllabus for the current academic year. All test papers and question banks for Class 10 Mathematics and CBSE Assignments for Mathematics Class 10 will be really helpful for standard 10th students to prepare for the class tests and school examinations. Class 10th students can easily free download in Pdf all printable practice worksheets given below.

Topicwise Assignments for Class 10 Mathematics Download in Pdf

More assignments for class 10 mathematics.

Advantages of Class 10 Mathematics Assignments

As we have the best and largest collection of Mathematics assignments for Grade 10, you will be able to easily get full list of solved important questions which can come in your examinations.
Students will be able to go through all important and critical topics given in your CBSE Mathematics textbooks for Class 10 .
All Mathematics assignments for Class 10 have been designed with answers. Students should solve them yourself and then compare with the solutions provided by us.
Class 10 Students studying in per CBSE, NCERT and KVS schools will be able to free download all Mathematics chapter wise worksheets and assignments for free in Pdf
Class 10 Mathematics question bank will help to improve subject understanding which will help to get better rank in exams

Frequently Asked Questions by Class 10 Mathematics students

At https://www.cbsencertsolutions.com, we have provided the biggest database of free assignments for Mathematics Class 10 which you can download in Pdf

We provide here Standard 10 Mathematics chapter-wise assignments which can be easily downloaded in Pdf format for free.

You can click on the links above and get assignments for Mathematics in Grade 10, all topic-wise question banks with solutions have been provided here. You can click on the links to download in Pdf.

We have provided here topic-wise Mathematics Grade 10 question banks, revision notes and questions for all difficult topics, and other study material.

We have provided the best collection of question bank and practice tests for Class 10 for all subjects. You can download them all and use them offline without the internet.

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NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

NCERT Solutions Class 10 Maths Chapter 2 Polynomials are provided here to help the students of CBSE class 10. Our expert teachers prepared all these solutions as per the latest CBSE syllabus and guidelines. In this chapter, we have discussed the Zeroes of polynomial, Relationship between Zeroes and Coefficients of a Polynomial and Division Algorithm for Polynomials in details. CBSE Class 10 Maths solutions provide a detailed and step-wise explanation of each answer to the questions given in the exercises of NCERT books.

CBSE Class 10 Maths Chapter 2 Polynomials Solutions

Below we have given the answers to all the questions present in Polynomials in our NCERT Solutions for Class 10 Maths chapter 2. In this lesson, students are introduced to a lot of important concepts that will be useful for those who wish to pursue mathematics as a subject in their future classes. Based on these solutions, students can prepare for their upcoming Board Exams. These solutions are helpful as the syllabus covered here follows NCERT guidelines.

NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.1

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.1

NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.2 00001

NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.3

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.3 00001

NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.4

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.4 00001

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Mathematics lab manual class 10, math assignment ch-2 class x | polynomials.

ASSIGNMENT | CHAPTER-2

(a) 4x² - 4x + 1 [Ans; 1/2],

$class 10 maths ch 2 assignment$

(d) 2x² + 5x + 2 [Ans -2, -1/2]

$class 10 maths ch 2 assignment$

(h) 9y² - 6y + 1 [Ans; 1/3, 1/3],

(i) 3x² - 5x - 2 [Ans; 2, -1/3]

$class 10 maths ch 2 assignment$

(a) 2x² + 2x + 3 [Ans; -1, 3/2],

(b) x² - 7x - 7 [Ans; 7, -7]

$class 10 maths ch 2 assignment$

[Ans a = 2]

[Ans k = 9]

[Ans; a = - 3]

$class 10 maths ch 2 assignment$

[Ans; k = - 16]

$class 10 maths ch 2 assignment$

Ans[x 2 + 4x + 4]

$class 10 maths ch 2 assignment$

Ans[k = - 3, - 1]

(a) P(x) = x⁴ + x² + 1; g(x) = x² + x + 1

Ans; q(x) = x² - x + 1

(b) P(y) = 4y⁴ - 10y³ - 10y² + 30y - 15; g(y) = 2y - 5

Ans; q(y) = 2x 3 - 5x + 5/2

Ans; q(x) = 2x³ + 2x² + x + 1

[Ans; a = 1, b= - 7]

Ans; 1, - 2

[Ans; All zeroes are -1, 3, -1/3]

[Ans: All zeroes are -10, -1, - 2]

$class 10 maths ch 2 assignment$

[Ans: - 1, - 1]

$class 10 maths ch 2 assignment$

[Ans: -1/2, 3, -2, -1]

$class 10 maths ch 2 assignment$

[Ans: q(x) = - x² - 1, r(x) = - 5x + 10]

[Ans: 2x³ - 8x + 3]

[Ans; 5]

[Ans; 14x - 10]

[ Ans; 6]

[Ans: - 2x + 3]

[Ans; 3x - 1]

Ans [a = 1 & b = 2]

$class 10 maths ch 2 assignment$

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Polynomials Class 10 Notes CBSE Maths Chapter 2 (Free PDF Download)

Revision Notes
Chapter 2 Polynomials

Exam-Focused Revision Notes for CBSE Class 10 Maths Chapter 2 - Polynomials

For young students, tenth grade plays a vital role in their entire schooling education. It will become the turning point for their further education. So, gaining knowledge is very important. Class 10 Maths Chapter 2 Notes professionally designed can be the perfect source of material to prepare for their exams. It has a downloading option from the official website. The material is prepared by well-experienced faculty with a full of practice questions.

Vedantu is a platform that provides free NCERT Book Solutions and other study materials for students. You can download the NCERT Solution for Class 10 Science to score more marks in the examinations.

Overview of Deleted Syllabus for CBSE Class 10 Maths Chapter 1 Real Numbers

Download cbse class 10 maths revision notes 2024-25 pdf.

Also, check CBSE Class 10 Maths revision notes for All chapters:

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Access Class 10 Maths Chapter 2 Polynomials

If \[\text{p}\left( \text{x} \right)\] is a polynomial in $\text{x}$, the degree of the polynomial \[\text{p}\left( \text{x} \right)\] is the largest power of $\text{x}$ in \[\text{p}\left( \text{x} \right)\].

Types of Polynomials:

A linear polynomial is a polynomial with degree one.

A quadratic polynomial is a polynomial with degree two.

A cubic polynomial is a polynomial with degree three.

Zeros of a Polynomial:

If \[\text{p}\left( \text{x} \right)\] is a polynomial in $\text{x}$ and $\text{k}$ is any real number, the value obtained by substituting $\text{k}$ for $\text{x}$ in \[\text{p}\left( \text{x} \right)\] is known as the value of \[\text{p}\left( \text{x} \right)\] when \[\text{x = k}\]and is denoted by \[\text{p}\left( \text{k} \right)\]. If \[\text{p}\left( \text{k} \right)\text{ = 0}\], a real number $\text{k}$ is said to be a zero of a polynomial \[\text{p}\left( \text{x} \right)\].

The Geometrical Meaning of Polynomial Zeros:

The equation \[\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c}\] can have three cases for the graphs

Case (i):

Here, the graph cuts \[\text{x-}\]axis at two distinct points \[\text{A}\]and \[\text{A }\!\!'\!\!\text{ }\].

$The graph cuts \[\text{x-}\]axis at two distinct points \[\text{A}\]and \[\text{A }\!\!'\!\!\text{ }\]$

Case (ii): Here, the graph cuts the \[\text{x-}\]axis at exactly one point.

$The graph cuts the \[\text{x-}\]axis at exactly one point$

Case (iii): Here, the graph is either completely above the \[\text{x-}\]axis or completely below the \[\text{x-}\]axis.

$the graph is either completely above the \[\text{x-}\]axis or completely below the \[\text{x-}\]axis$

If \[\text{ }\!\!\alpha\!\!\text{ }\] and \[\text{ }\!\!\beta\!\!\text{ }\] are the zeroes of the quadratic polynomial\[\text{p}\left( \text{x} \right)\text{ = a}{{\text{x}}^{\text{2 }}}\text{+ bx + c, a}\ne \text{0}\], then it is known that \[\text{x - }\!\!\alpha\!\!\text{ }\] and \[\text{x - }\!\!\beta\!\!\text{ }\] are the factors of \[\text{p}\left( \text{x} \right)\].

\[\text{ }\!\!A\!\!\text{ + }\!\!\beta\!\!\text{ = - }\dfrac{\text{b}}{\text{a}}\]

\[\text{ }\!\!\alpha\!\!\text{ }\!\!\beta\!\!\text{ = }\dfrac{\text{c}}{\text{a}}\]

Division Algorithm for Polynomials:

If \[\text{p}\left( \text{x} \right)\] and \[\text{g}\left( \text{x} \right)\] are any two polynomials with \[\text{g}\left( \text{x} \right)\ne 0\], then polynomials \[\text{q}\left( \text{x} \right)\] and \[\text{r}\left( \text{x} \right)\] can be found such that \[\text{p}\left( \text{x} \right)\text{= g}\left( \text{x} \right)\text{ }\!\!\times\!\!\text{ q}\left( \text{x} \right)\text{ + r(x)}\], where \[\text{r}\left( \text{x} \right)=0\] or degree of \[\text{r}\left( \text{x} \right)<\] degree of \[\text{g}\left( \text{x} \right)\].

This result is known as the Division Algorithm for polynomials.

An example would make it easier to understand. So, consider a cubic polynomial x 3 - 3x 2 - x + 3.

Assuming that one of its zeroes is $1$, it is clear that \[\text{x - 1}\] is a factor of x 3 - 3x 2 - x + 3.

So, x 3 - 3x 2 - x + 3 can be divided by x -1 Taking out this factor, (x - 1)(x 2 - 2x - 3).

Next, get the factors of x 2 - 2x - 3 by splitting the middle term. (x + 1)(x - 3).

x 3 −3x 2 −x+3=(x−1)(x+1)(x−3)

So, all the three zeroes of the cubic polynomial are $1, − 1, 3$.

Mastering Polynomials: Comprehensive Class 10 Notes for Polynomial - A Chapter Overview

Class 10 Maths Chapter 2 Notes is available in a PDF format on the main website of Vedantu. Students can avail of this opportunity to take a physical copy, which avoids internet issues. This pdf is also helpful to store for the future. It is also useful to refer during the time of entrance test, olympiads, and other examinations.

Introduction:-

The Notes of ch 2 Maths Class 10 has started the lesson similar to all the chapters with an introduction. In this introduction part, the notes have provided a quick recap of all the previous knowledge students have gained in the earlier classes. A polynomial is an equation with one variable and changes its type based on the variable's degree. If the variable has a single degree, then it is called the linear polynomial. If the variable's degree is 2, then the polynomial is set to be quadratic polynomial. Similarly, the cubic polynomial is the polynomial that has a degree of 3. It is also explained that the value of the polynomial and if the value is 0, then it is called 0 polynomial.

CBSE Class 10 Math Revision Notes

Revision Notes For CBSE Class 10 Math Chapter 1 Real Numbers

Revision Notes For CBSE Class 10 Math Chapter 2 Polynomials

Revision Notes For CBSE Class 10 Math Chapter 3 Pair of Linear Equations In Two Variables

Revision Notes For CBSE Class 10 Math Chapter 4 Quadratic Equations

Revision Notes For CBSE Class 10 Math Chapter 5 Arithmetic Progression

Revision Notes For CBSE Class 10 Math Chapter 6 Triangles

Revision Notes For CBSE Class 10 Math Chapter 7 Coordinate Geometry

Revision Notes For CBSE Class 10 Math Chapter 8 Introduction To Trigonometry

Revision Notes For CBSE Class 10 Math Chapter 9 Some Application of Trigonometry

Revision Notes For CBSE Class 10 Math Chapter 10 Circles

Revision Notes For CBSE Class 10 Math Chapter 11 Constructions

Revision Notes For CBSE Class 10 Math Chapter 12 Area Related To Circles

Revision Notes For CBSE Class 10 Math Chapter 13 Surface Area And Volume

Revision Notes For CBSE Class 10 Math Chapter 14 Statistics

Revision Notes For CBSE Class 10 Math Chapter 15 Probability

Geometrical Meaning of The Zeros of A Polynomial

As it is already clear that the zero polynomial is a polynomial whose value is zero. In addition to the ordinary polynomials, the zero polynomial is very important when compared to the other. Because while plotting a graph for the zero quadratic polynomial, we have a result of the parabola. This parabola also appears at different points, which represent different cases. Those cases are specified by Notes of Chapter Polynomials Class 10.

One parabola may cut the x-axis and y-axis at two points.

When one parabola may cut the x-axis and y-axis at a single point.

The parabola may be entirely above the x-axis or completely below the x-axis.

One parabola may be completely above the y-axis or completely below the y-axis.

These are the various cases observed in the zero polynomial while plotting graphs are explained with several solved examples by taking different values.

Relationship Between Zeros And Coefficients of a Polynomial

In this section, the Notes of Chapter 2 Maths Class 10 explains the questions of a polynomial by reminding the factorization method, which had been learned in the previous class. In the factorization, the quadratic polynomial has been split into multiple factors, which means writing each term separately. Then taking a common term and obtaining the result. The remaining two terms can be treated as factors.

For an instance,

3x 2 +14x+ 8 is the polynomial.

Then split the equation into four terms without changing the values.

3x 2 + 2x +12x +8 = 0

Then take common and split into two factors.

(3x+2)(x+4)=0

Hence the factors are, -⅔, -4.

Division Algorithm For Polynomials

After understanding the questions and factors, the Class 10 Maths ch 2 Notes notes the division algorithm concerning polynomials. So far, the PDF has discussed quadratic polynomials. But to understand the division algorithm, Class 10 Maths Polynomials Notes specifies that we need to consider cubic polynomials with three zeros. By taking a single value, we need to learn the finding process of the other two values.

To get the first term of coefficient, we need to divide the highest degree term of dividend.

The same process has to be followed for the next term also.

If the degree of the divisor is less than the degree of dividend, then we need to use a formula to find the quotient.

Dividend = divisor * quotient + Remainder.

This is the process and formula for division algorithm explained by Chapter 2 Maths Class 10 Notes

FAQs on Polynomials Class 10 Notes CBSE Maths Chapter 2 (Free PDF Download)

1. Find the value of “p” from the polynomial x 2 + 3x + p, if one of the zeroes of the polynomial is 2.

It is given that, 2 is the zero of the polynomial.

We know that if α is a zero of the polynomial p(x), then p(α) = 0

So,Substituting x = 2 in x2 + 3x + p,

We get, 22 + 3(2) + p = 0

4 + 6 + p = 0

10 + p = 0

p = -10.

Hence the value of p is -10.

2. How many zeros do the polynomial (x – 3) 2 – 4 can have? Also, find them.

Given the polynomial equation is (x – 3) 2 – 4

Now, we need to expand this equation.

x 2 + 9 – 6x – 4

x 2 – 6x + 5

As the equation has a degree of 2, it is called a quadratic polynomial. So, the number of zeroes will be 2.

Now, solve x 2 - 6x + 5 = 0 to get the factors by using the factorization method.

So, x 2 – x – 5x + 5 = 0

x(x – 1) -5(x – 1) = 0

(x – 1)(x – 5) = 0

x = 1, x = 5

So, the factors are 1 and 5.

These are the two zeros that we need to find.

Hence it is solved.

3. What is polynomial in the context of Chapter 2 Class 10?

The term Polynomial comes from the word 'poly' which means 'many' and 'nominal' which means term. Therefore, polynomials refer to many terms. A polynomial is made up of many terms which can only be substrated, added, or multiplied. A polynomial with the highest exponent of variables is known as the degree of the polynomial. The polynomial with degree one is known as a linear, degree two is known as a quadratic, and degree three is known as a cubic polynomial.

4. What are important polynomial notes for chapter 2 class 10?

While studying Chapter 2 Class 10 Maths , it is important for you to understand the concept of the polynomial. This is an important chapter from your board examination point of view. The important polynomial notes include types of polynomials, the degree of polynomials, zeros of a polynomial, formulas, and all the algorithms. These notes will help you during your revision time as well, to help you score well during your board exam.

5. Where can I find NCERT Solutions for Class 10 Maths Chapter 2?

Vedantu provides students with the best NCERT Solutions for the Class 10 Maths Chapter 2. It provides all important questions and solutions for Chapter 2 “Polynomials''. It covers all types of questions from the board examination point of view. They have a variety of questions to help students understand the chapters and important concepts better. These solutions and questions are extremely important for all CBSE students from the viewpoint of examinations. Also, the solutions PDF is available for free download on the Vedantu mobile app.

6. How can you divide one polynomial with another polynomial?

A polynomial refers to an expression with more than two algebraic terms. It refers to the sum of many terms that have different powers of the same variable. You can divide one polynomial with another polynomial using two methods namely long division, and synthetic division. The easiest way to divide one polynomial with another polynomial is using long division. Using long division will help to test whether or not a single polynomial has another one as a factor.

7. How to ace Class 10 Maths Chapter 2?

Class 10 Maths Chapter 2 is an important topic from the examination point of view. Students can rely on the study materials available on the official website of Vedantu or their app and they are available free of cost. One of the most important ways to ace Chapter 2 is to complete all the NCERT questions and answers and carry out thorough revision. Revision is an important part of exam preparation. By completing solutions and exercises the students can ace their Class 10 Maths Chapter 2.

NCERT Solutions For Class 10 Maths Chapter 2 Polynomials

Here are the NCERT Solutions for Class 10 Maths Chapter 2 Polynomials to help students prepare for their exams. These solutions were written by Math topic experts to help students prepare for their first-term exams. They approach these issues in such a way that students can practice the questions from Chapter 2 Polynomials using the NCERT Solutions. Students will find it easy to understand these Maths NCERT Class 10 Solutions because they include step-by-step explanations.

NCERT Solutions for Class 10 Maths would be a great study resource for students. Students who complete these Polynomials NCERT Solutions for Class 10 Maths will perform well on their first and second semester exams. Special attention is made to following the new CBSE Syllabus for 2024-25 when generating these solutions.

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This is one of the most important topics in math, hence it belongs under the Algebra unit of the Class 10 Maths CBSE Term I exams, which has a 20-mark weighted. One question is the typical number of questions posed from this chapter.

Polynomials: An Overview

Relationship between Zeros and Coefficients of a Polynomial Division Algorithm for Polynomials has a geometrical meaning.

After discussing polynomials in one variable and their degrees in the previous lesson, Lesson 10 delves deeper into polynomials. The answers to a range of questions about polynomials and their applications are covered in this chapter of NCERT Solutions for Class 10 Maths. We look into the relationship between the zeros of quadratic polynomials and their coefficients, as well as the division technique for polynomials of integers.

Polynomials are introduced in section 2.1 of the chapter, followed by parts 2.2 and 2.3, which address two extremely important topics.

Geometrical Meaning of a Polynomial's Zeroes - There is one question with six different instances.

Relationship between a polynomial's zeros and coefficients – Examine the relationship between a quadratic polynomial's zeroes and coefficients by solving two problems in Exercise 2.2, each of which has six parts.

NCERT Solutions Class 10 Maths All Chapters:

Ncert solutions for class 10 maths chapter 2: polynomials: key features:.

It covers the new CBSE Class 10 Maths term-by-term syllabus for 2024-25.
You will be confident in your ability to achieve well in exams after studying these NCERT Solutions prepared by our subject specialists.
It follows NCERT guidelines, which aid in students' thorough preparation.
It contains all of the important questions from the standpoint of the examination.

Here are the notes for CBSE Class 10 Maths Chapter 2 Polynomial . With several examples, we will cover everything from what is a polynomial and its kinds to algebraic expressions, degree of a polynomial expression, graphical representation of polynomial equations, factorization, and the link between zeroes and the coefficient of a polynomial.

Algebraic Expression

An algebraic expression is a formula that contains variables, constants, and mathematical operators.

A sum of terms, often known as building blocks for expressions, is an algebraic expression.

Variables and constants are combined to form a term. In some cases, a term can be an algebraic expression in and of itself.

- 3 is an example of a phrase that is simply a constant.

- 2x is the product of the constant '2' and the variable 'x'.

- 4xy is the product of the constant '4' and the variables 'x' and 'y'.

- 5x2y, which is the sum of the numbers 5, x, x, and y.

The coefficient refers to the constant in each expression.

3x2y+4xy+5x+6 is an algebraic expression that is the sum of four terms: 3x2y, 4xy, 5xy, and 6.

Exponents in algebraic expressions can be rational values. A polynomial, on the other hand, is an algebraic expression in which the exponent on any variable is a whole integer; an example of a polynomial is 5x3+3x+1. It's also an algebraic expression.

2x+3x is a polynomial, however it is not an algebraic expression. because x has an exponent of 1/2, which is not a whole number.

Degree of a Polynomial

The degree of a polynomial in one variable is equal to the largest exponent on the variable in the polynomial.

Example: The polynomial x2+2x+3 has a degree of 2 since the largest power of x in the provided expression is x2.

Polynomials classified by the number of terms

A one term polynomial is monomial . 2x, 6x2, 9xy, etc.

Binomial – A two-term polynomial. 4x2+x, 5x+4 are two examples.

A trinomial is a polynomial with three terms. x2+3x+4 is an example.

Degree Polynomials Types: -

A linear polynomial is a polynomial that has only one degree.

A linear polynomial, for example, is 2x+1.

Quadratic Polynomial is a type of polynomial that has four coefficients

A quadratic polynomial is a polynomial with two degrees.

3x2+8x+5 is a quadratic polynomial, for example.

NCERT Solutions for class 10 Maths Chapter 2- Polynomials Exercise 2.1

NCERT Solutions of Chapter 2 for Class 10 Maths have been solved here with detailed explanation by our specialists of the particular subject. These solutions are students’ friendly solutions as these are directly linked from the concepts students last studied in the previous grade.

All the NCERT guidelines are considered strongly by our specialists while preparing solutions. The solutions are checked by our professionals after our specialists complete them so as to make the solutions error free. These solutions work on building the concepts of students so that they never get any confusion in these particular topics.

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Exercise 2.1 Polynomials Class 10 Maths, covers the concept of zeroes of a polynomial. This section also includes the geometrical representation and meaning of linear and quadratic polynomial s.

One liner explanation of the concept’s students will learn in this particular of NCERT chapter 2.

Zeroes of a polynomials may be defined as the points on which the value of polynomial becomes zero; for example, let us assume a polynomial p(x) and at any point k value of p(x) is zero, i.e., p(k)=0. Thus, k is the zero of polynomial p(x). Geometrically, we can say that at point k, the polynomial will cut the x- axis.

Chapter 2 - Polynomials NCERT Solutions for Class 10 Maths:

The first exercise in Chapter 2 of Class 10 Maths is Exercise 2.1 which continues from where we left the concept of polynomials in Class 9. Polynomials are introduced in Class 9 and examined in further depth in Class 10 by looking at various situations of the geometrical significance of a polynomial's zeroes.

Key Benefits of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.1:

You can use these NCERT Solutions to solve and revise NCERT exercise 2.1 questions.
It will build your concepts as well as it will boost up your confidence which you need, to take the Class 10 CBSE 2025 exams.
It explains the concept of polynomials in the detailed manner and to understand better it is important to solve each question given in the NCERT textbook exercises of the chapter.
It is very important to understand the idea of polynomials and their zeroes with the use of graphs. Zeroes of the polynomials, also known as roots of the polynomials are very essential as these are the solutions for any given polynomial for which the unknown variable's value must be determined. If we know the roots, we can evaluate the value of a polynomial to zero.

Frequently Asked Questions on NCERT Solutions of Chapter 2 Exercise 2.1:

What is the weightage of this Chapter 2 Exercise 2.1 in CBSE examination?

Answer: Chapter 2 Exercise 2.1 of Class 10 Maths is very important from examination point of view as well. It will constitute approximately 4 marks of the total marks of the exam paper.

Is it important to solve NCERT solutions?

Answer: Yes, it is very important to solve NCERT questions as it clears the concepts of the questions which will come in very handy even if the examiner twists the language or the pattern of the basic questions.

What is the importance of Chapter 2 Exercise 2.1 of Class 10?

Answer: Chapter 2 Exercise 2.1 of Class 10 is considered as the most significant chapter in Class 10 Math because you will gain a proper understanding of the polynomial equations which will help you in later chapters. Understanding the graph and connecting it to the equation is fundamental as you will be asked similar questions in subsequent chapters. As a result, try to attempt answer as many questions as possible in order to gain a thorough comprehension of the subject.

NCERT Solutions for class 10 Maths Chapter 2- Polynomials Exercise 2.2

NCERT Solutions for Class 10 Maths, Chapter 2 Exercise 2.2 , have been provided here with full explanations by our subject experts. These are student-friendly solutions since they are directly related to subjects that students previously covered in the preceding grade. When preparing solutions, our experts take into account all of the NCERT requirements .

Our professionals evaluate the solutions once our specialists finish them to ensure that they are error-free. These solutions help students to develop their notions so that they are never confused about these concepts. Students can use these NCERT chapter-by-chapter solutions to study and prepare for their CBSE 2025 exams. NCERT Exercise 2.2 Chapter 2- polynomials Solutions are presented in a step-by-step and easy-to-understand manner. Once you've gone over these NCERT solutions for Class 10 Maths, you'll be able to solve them quickly.

NCERT Exercise 2.2 Chapter 2- polynomials, illustrate the relationship between the zeroes of a polynomial with the coefficient of polynomials.

Brief explanation of exercise 2.2 class 10 Maths : The relationship of zeroes of polynomial with coefficients of polynomials. In this exercise Linear polynomial and quadratic polynomials are discussed. We already know that zeroes of a linear polynomial can be calculated by simply putting the linear polynomial, p(x)= ax+b equal to zero and find the value of x which will turn out to be -b/a.

Now, quadratic polynomial having degree 2 and it will relate the sum of the two roots and product of the two roots with the coefficient of quadratic polynomial.

The two roots are represented by two Greek symbols α and β.

Chapter 2 - Polynomials NCERT Solutions for Class 10 Maths: The second exercise in Chapter 2 of Class 10 Maths is Exercise 2.2 solely explains how to find sum of roots of a polynomial and product of roots of polynomial without actually finding the roots of the equation. It is done by using the relationship between the sum of roots and the product of roots with the coefficient of polynomials respectively.

NCERT Solutions for Class 10 Maths Chapter 2- Polynomials: Key Benefits

These NCERT Solutions will assist you in answering and revising all of the problems in Exercise 2.1.
It adheres to NCERT criteria, which aid in the proper preparation of students.
From the standpoint of the examination, it comprises all of the key questions and solving these NCERT questions will help you to do better in the examination.

Frequently Asked Questions:

Answer: Yes, solving NCERT questions is very significant since it clarifies the principles of the questions, which will come in helpful even if the examiner twists the language or pattern of the core questions.

What is the weightage of this Chapter 2 in CBSE examination?

Answer: Chapter 2 of Class 10 Maths is also highly crucial in terms of examinations. It will account for about 4 percent of the total marks on the exam paper.

Is it important to solve examples before the exercise question?

Answer: Yes, it is important to solve the examples as well as these are the basis of the concepts as well as the questions in the further exercise. NCERT examples are designed to assist students in a timely manner in the event of questions or uncertainties. The answers are thoroughly presented, with examples that are both relevant and understandable.

What is the summary of exercise 2.2 Chapter 2 polynomials?

Answer: This exercise states that the sum of zeroes of an equation is equal to - (coefficient of x) divided by the coefficient of square of x. The zero product is equal to the constant term divided by the coefficient of square of x. The concept is taught using examples and the practice questions from the textbook.

NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.3

NCERT Solutions for 10 Maths Exercise 2.3 Chapter 2: Our topic specialists have ready the polynomials that you're going to notice here. These solutions area unit written by NCERT topic specialists so evaluated by them to make sure that students perceive the thought. This guarantees that the NCERT Solutions area unit easy enough for students of CBSE class 10 to grasp.

In Chapter two Polynomials class 10 Maths NCERT Exercise 2.3, The Division algorithmic rule for polynomials is mentioned. These answers area unit written with the aim of jutting to all or any NCERT rules. It conjointly makes bound that the complete topic is discussed while making an attempt to unravel issues, guaranteeing that the learner doesn't miss any topics.

What's the Division algorithmic rule for polynomials?

Division is also a computation throughout that things unit of measurement divided into equal elements. it's conjointly referred to as the multiplication operation's inverse.

For example, at intervals the multiplication, suppose that their unit of measurement a combine of groups of vi that make the twelve and presently if we have a tendency to tend to divide the twelve into a combine of groups there will be vi objects in each cluster. Here the twelve is presently the dividend, a combine of is that the divisor and vi is that the quotient. The Division algorithm for polynomials tells us that, if p(x) and g(x) unit of measurement the two polynomials, where g(x)≠0, we are going to write the division of polynomials as:

p(x) = q(x) * g(x) + r(x)

In this equation the p(x) denotes the dividend, q(x) denotes the quotient, g(x) denotes the divisor, r(x) denotes the rest.

Basics of Division

A dividend is also a full vary or the number of one thing that possesses to be split into equal components. Addition, subtraction, multiplication, and division unit of measurement the four basic operations on numbers. The division of variety into equal components is also a technique that leaves a remainder if the provided variety cannot be divided equally into components. thus, the division with a remainder contains the followings

* Dividend - Dividend is that the variety that's to be divided by the divisor.

* Divisor - the amount by that the dividend is to be divided is termed the divisor.

* Quotient - The resultant of the division is termed the quotient.

* Remainder - the amount that's left once division is termed the rest.

Polynomial division includes dividing one polynomial by a monomial, binomial, trinomial, or lower degree polynomial. The dividend's degree is larger than the divisor in associate passing polynomial division. to look at the result, multiply the divisor and quotient, then add the remaining, if any, to the result.

To divide a polynomial by another polynomial, follow these steps.

Step 1: organize the dividend and also the divisor so as of decreasing degrees.

Step 2: Divide the best degree term of the dividend by the best degree term of the divisor to induce the primary term of the quotient.

Step 3: Divide the best degree term of the new dividend made in step two by the biggest degree term of the divisor to induce the second term of the quotient.

Step 4: Repeat the steps once more till the degree of the rest is a smaller amount than the degree of the divisor.

To execute polynomial division, we'll use the division algorithmic rule stages.

Step one : The polynomials area unit already organized within the dropping order of their degrees.

Step 2: the primary term of the quotient is obtained by dividing the biggest degree term of the dividend with the biggest degree term of the divisor. initial Term = 2(x)*3/x=2(x)*2

Step 3: The new dividend is (x)*2+4(x)

Step 4 : The second term of the quotient is obtained by dividing the biggest degree term of the new dividend obtained in step two with the biggest degree term of the divisor. Second Term = (x)*2/x=x

Step 5 : The new dividend is 3(x)+3

Step 6 : The third term of the quotient is obtained by dividing the biggest degree term of the new dividend obtained in step four with the biggest degree term of the divisor. Third Term = 3(x)/x=3

Step 7: The quotient is 2(x)*2+2(x)+3

p(x) = 2(x)*3+3(x)*2+4(x)+3 is that the dividend

q(x) = 2(x)*2+x+3 is that the quotient

g(x) = x+1 is that the divisor

r(x) = zero is that the remainder

NCERT Solutions for Class 10 Maths Chapter 2- Polynomials– Key Features:

These NCERT Solutions can assist you in resolution and rewriting all of the queries during this specific exercise 2.3, permitting you to possess a deeper understanding of the concept.

Once going over the gradual answers provided by our topic specialists, you may be able to get high grades.
It aids in achieving smart results on the Maths tests.
It adheres to NCERT criteria, that aid within the correct preparation of pupils.
From the stance of the check, NCERT solutions Class 10 contains all of the key queries.

NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4

This Chapter 2 Polynomials NCERT Solutions for Class 10 Math Students can use Exercise 2.4 to help them with their academics. These solutions were created by our Maths subject expert to help you prepare for the Class 10 2025 exams. These specialists create NCERT Math Solutions to make it easier for students to tackle NCERT questions quickly. They also consider how easy it is for you to grasp the topic and how quickly they can learn from it.

This 2.4 Math Exercise is optional, and it is not provided for the final examination point of view but it is important for you to understand many significant concepts later in Maths.

This NCERT Maths Solution for Class 10 , comprises additional chapter-related questions to practice. Each response to the questions is given a full solution by our specialists. It also covers the whole syllabus of NCERT Class 10th Maths book . These will help you to score high in exams and it will also help to revise the NCERT complete syllabus.

Main Advantages of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4:

Students will be able to obtain higher grades by practicing step-by-step these NCERT solutions provided by our subject specialist teachers.
It completely contains NCERT criteria , which benefit the students in complete preparation for exams.
It covers the whole syllabus and also consists of all of the key questions.
It will help the students to score well in the first-term Math examinations.
These NCERT Solutions can also be used to solve and review all of the problems in exercise 2.4.

Why Choose Us?

We make Mathematics interesting for even those students who are scared even by the name of this subject. Highly skilled and qualified teachers have created the subject of this solution. These solutions are the best if you want to get your concepts cleared easily at the primary stage.

You don’t need to go to Maths tuition every day, all you need to do is practice these NCERT Solution papers well and it will help you to score high in the examination.

If you go through these NCERT Solution papers and practice Maths religiously, you will get to see the difference in yourself in your next examination. You don’t need to practice Maths for hours anymore. All you need to do is to go through these NCERT Maths Class 10 questions and solutions provided here. There is no important question on the Maths Chapter 2 Class 10 that we have missed in these notes.

With these NCERT Solutions for Class 10 Math , you will also learn how to solve a problem related to the Polynomial Chapter in the easiest manner possible. Once you learn it, no one can stop you from getting high marks in Maths.

This time with these notes, you will be surely able to surprise your teachers as well as your parents. Study the Exercise 2.4 Class 10 Maths NCERT Solutions for free to score better in your exams.

CBSE Sample Papers 2022-23 for Class 10 | For 2023 Board Exams

FAQs (Frequently Asked Questions):

What is the best way to score full marks in NCERT Exercise 2.4 of Chapter 2 of Class 10 Maths?

Students of Class 10 can score full marks in Exercise 2.4 of Chapter 2 by practicing all questions given in the Exercise. The solutions are prepared by expert subject teachers for helping the students in concept understanding and scoring maximum marks.

What does NCERT Solutions Class 10th Maths Chapter 2 Exercise 2.4 deal with?

These NCERT Solutions for Class 10 Maths Chapter 2 , Exercise 2.4 will help you in understanding the concept of Polynomials in a clear manner. This chapter helps students to understand many other significant concepts in Maths.

Studying from NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths is sufficient to prepare for the exams?

NCERT Solutions for Chapter 2 of Class 10 Maths are sufficient for the students to prepare for the exams as they can solve all NCERT solutions by practicing these solutions daily. The solutions are prepared by expert teachers by keeping this point in mind that students can score high in the examinations. Students can easily understand the pattern of the questions asked in the examinations and this can help them to solve all questions easily in the exams in a limited given time.

Frequently Asked Questions: NCERT Solutions for Class 10 Maths Chapter 2:

Where can I find the proper NCERT Solutions for Chapter 2 of Class 10 Maths?

The NCERT Textbook Solutions for Polynomials have been meticulously constructed. All of these solutions are based on the new CBSE pattern, ensuring that students comprehend the topic thoroughly in preparation for their exams.

Is it compulsory to finish all of the problems in NCERT Solutions for Class 10 Maths Chapter 2?

Yes. Because they are the most important questions in the exam. These questions have been answered by experts to help students complete the work quickly. These solutions help students have a better understanding of polynomials.

What topics are included in the NCERT Solutions for Class 10 Maths Polynomials?

The concepts are covered in the NCERT Solutions for Class 10 Maths. Polynomial division algorithm, geometrical meaning of polynomial zeros, relationship between zeros and coefficients of a polynomial, and introduction to polynomials If students are able to answer polynomial problems, they will be successful.

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NCERT Exemplar
Maths Exemplar Class 10
Polynomials

NCERT Exemplar Class 10 Maths Solutions for Chapter 2 - Polynomials

Ncert exemplar solutions class 10 maths chapter 2 – free pdf download.

NCERT Exemplar Class 10 Maths Chapter 2 Polynomials are provided here for students to prepare for the board exam. These solutions are prepared as per NCERT guidelines and the latest CBSE syllabus (2023-2024) by our subject experts. With the help of exemplar problems and solutions, students will be able to revise the complete chapter and score better marks in the exam.

In this chapter, students will learn to solve exemplar problems based on polynomials and their subtopics like:

Geometrical Meaning of the Zeroes of a Polynomial
Division Algorithm for Polynomial
Zeroes and Coefficients of a Polynomial and Their Relationships

In NCERT Exemplar Class 10 Maths Chapter 2, students will learn about the concept of polynomials which has been discussed in detail. They will also study important topics like the relationship between coefficients and zeroes, the division algorithm for polynomials and the geometrical meaning of the zeroes of a polynomial . As students need to be thorough with these topics, free NCERT Exemplar for Chapter 2 – Polynomials is provided here.

Students can download the Class 10 Maths Chapter 2 NCERT Exemplar PDF from the link provided below.

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Exercise 2.1.

Choose the correct answer from the given four options in the following questions:

1. If one of the zeroes of the quadratic polynomial ( k– 1) x 2 + k x + 1 is –3, then the value of k is

(A) 4/3 (B) -4/3

2/3 (D) -2/3

Explanation:

According to the question,

-3 is one of the zeros of quadratic polynomial (k-1)x 2 +kx+1

Substituting -3 in the given polynomial,

(k-1)(-3)²+k(-3)+1=0

(k-1)9+k(-3)+1 = 0

9k-9-3k+1=0

Therefore, k=4/3

Hence, option (A) is the correct answer.

2. A quadratic polynomial, whose zeroes are –3 and 4, is

(A) x 2 – x + 12 (B) x 2 + x + 12

Sum of zeroes, α+ β= -3+4 =1

Product of Zeroes, αβ = -3× 4 = -12

Therefore, the quadratic polynomial becomes,

x²- (sum of zeroes)x+(product of zeroes)

= x²- (α+ β)x+(αβ)

= x² – (1)x + (-12)

= x² – x -12

divide by 2, we get

= x²/2 – x/2 -12/2

= x²/2 – x/2 -6

Hence, option (C) is the correct answer.

3. If the zeroes of the quadratic polynomial x 2 + ( a + 1) x + b are 2 and –3, then

(A) a = –7, b = –1 (B) a = 5, b = –1

(D) a = 2, b = – 6

x² + (a+1)x + b

Given that, the zeroes of the polynomial = 2 and -3,

2² + (a+1)(2) + b = 0

4 + 2a+2 + b = 0

6 + 2a+b = 0

2a+b = -6 —– (1)

When x = -3,

(-3)² + (a+1)(-3) + b = 0

9 – 3a-3 + b = 0

6 – 3a+b = 0

-3a+b = -6 —– (2)

Subtracting equation (2) from (1)

2a+b – (-3a+b) = -6-(-6)

2a+b+3a-b = -6+6

Substituting the value of ‘a’ in equation (1), we get,

2a + b = -6

2(0) +b = -6

Hence, option (D) is the correct answer.

4. The number of polynomials having zeroes as –2 and 5 is

(A) 1 (B) 2

(D) more than 3

The zeroes of the polynomials = -2 and 5

We know that the polynomial is of the form,

p(x) = ax 2 + bx + c.

Sum of the zeroes = – (coefficient of x) ÷ coefficient of x 2 i.e.

Sum of the zeroes = – b/a

– 2 + 5 = – b/a

3 = – b/a

b = – 3 and a = 1

Product of the zeroes = constant term ÷ coefficient of x 2 i.e.

Product of zeroes = c/a

(- 2)5 = c/a

– 10 = c

Substituting the values of a, b and c in the polynomial p(x) = ax 2 + bx + c.

We get, x 2 – 3x – 10

Therefore, we can conclude that x can take any value.

5. Given that one of the zeroes of the cubic polynomial ax 3 + bx 2 + cx + d is zero, the product of the other two zeroes is

(A) (–c/a) (B) c/a

We have the polynomial,

ax 3 + bx 2 + cx + d

We know that,

Sum of product of roots of a cubic equation is given by c/a

It is given that one root = 0

Now, let the other roots be α, β

So, we get,

αβ + β(0) + (0)α = c/a

Hence the product of other two roots is c/a

Hence, option (B) is the correct answer

Exercise 2.2

1. Answer the following and justify:

(i) Can x 2 – 1 be the quotient on division of x 6 + 2 x 3 + x – 1 by a polynomial in x of degree 5?

No, x 2 – 1 cannot be the quotient on division of x 6 + 2x 3 + x – 1 by a polynomial in x of degree 5.

Justification:

When a degree 6 polynomial is divided by degree 5 polynomial,

The quotient will be of degree 1.

Assume that (x 2 – 1) divides the degree 6 polynomial with and the quotient obtained is degree 5 polynomial (1)

According to our assumption,

= (degree 7 polynomial)

From the above equation, it is clear that, our assumption is contradicted.

x 2 – 1 cannot be the quotient on division of x 6 + 2x 3 + x – 1 by a polynomial in x of degree 5

Hence Proved.

(ii) What will the quotient and remainder be on division of ax 2 + bx + c by px 3 + qx 2 + rx + s , p ≠ 0?

Degree of the polynomial px 3 + qx 2 + rx + s is 3

Degree of the polynomial ax 2 + bx + c is 2

Here, degree of px 3 + qx 2 + rx + s is greater than degree of the ax 2 + bx + c

Therefore, the quotient would be zero,

And the remainder would be the dividend = ax 2 + bx + c.

(iii) If on division of a polynomial p ( x ) by a polynomial g ( x ), the quotient is zero, what is the relation between the degrees of p ( x ) and g ( x )?

p(x)= g(x) × q(x)+r(x)

When q(x)=0, then r(x) is also = 0

So, now when we divide p(x) by g(x),

Then p(x) should be equal to zero

Hence, the relation between the degrees of p (x) and g (x) is the degree p(x)<degree g(x)

(iv) If on division of a non-zero polynomial p ( x ) by a polynomial g ( x ), the remainder is zero, what is the relation between the degrees of p ( x ) and g ( x )?

In order to divide p(x) by g(x)

Degree of p(x) > degree of g(x)

Degree of p(x)= degree of g(x)

Therefore, we can say that,

The relation between the degrees of p ( x ) and g ( x ) is degree of p(x) > degree of g(x)

(v) Can the quadratic polynomial x 2 + kx + k have equal zeroes for some odd integer k > 1?

A Quadratic Equation will have equal roots if it satisfies the condition:

b² – 4ac = 0

Given equation is x² + kx + k = 0

a = 1, b = k, x = k

Substituting in the equation we get,

k² – 4 ( 1 ) ( k ) = 0

k² – 4k = 0

k ( k – 4 ) = 0

k = 0 , k = 4

But in the question, it is given that k is greater than 1.

Hence the value of k is 4 if the equation has common roots.

Hence if the value of k = 4, then the equation ( x² + kx + k ) will have equal roots.

Exercise 2.3

Find the zeroes of the following polynomials by factorisation method.

1. 4 x 2 – 3 x – 1

4 x 2 – 3 x – 1

Splitting the middle term, we get,

4x 2 -4x+1x-1

Taking the common factors out, we get,

4x(x-1) +1(x-1)

On grouping, we get,

(4x+1)(x-1)

So, the zeroes are,

4x+1= 0⇒ 4x=-1 ⇒x= (-1/4)

(x-1) = 0 ⇒ x=1

Therefore, zeroes are (-1/4) and 1

Verification:

Sum of the zeroes = – (coefficient of x) ÷ coefficient of x 2

α + β = – b/a

1 – 1/4 = – (- 3)/4 = ¾

Product of the zeroes = constant term ÷ coefficient of x 2

1(- 1/4) = – ¼

– 1/4 = – 1/4

2. 3 x 2 + 4 x – 4

3x 2 + 4x – 4

3x 2 + 6x – 2x – 4

3x(x+2) -2(x+2)

(x+2)(3x-2)

x+2=0 ⇒ x= -2

3x-2=0⇒ 3x=2⇒x=2/3

Therefore, zeroes are (2/3) and -2

– 2 + (2/3) = – (4)/3

= – 4/3 = – 4/3

Product of the zeroes = (- 2) (2/3) = – 4/3

3. 5 t 2 + 12 t + 7

5t 2 + 12t + 7

5t 2 +5t + 7t + 7

5t (t+1) +7(t+1)

(t+1)(5t+7)

t+1=0 ⇒ y= -1

5t+7=0 ⇒ 5t=-7⇒t=-7/5

Therefore, zeroes are (-7/5) and -1

(- 1) + (- 7/5) = – (12)/5

= – 12/5 = – 12/5

(- 1)(- 7/5) = 7/5

4. t 3 – 2 t 2 – 15 t

t 3 – 2 t 2 – 15 t

Taking t common, we get,

t ( t 2 -2t -15)

Splitting the middle term of the equation t 2 -2t -15, we get,

t( t 2 -5t + 3t -15)

t (t (t-5) +3(t-5)

t (t+3)(t-5)

t+3=0 ⇒ t= -3

t -5=0 ⇒ t=5

Therefore, zeroes are 0, 5 and -3

Sum of the zeroes = – (coefficient of x 2 ) ÷ coefficient of x 3

α + β + γ = – b/a

(0) + (- 3) + (5) = – (- 2)/1

Sum of the products of two zeroes at a time = coefficient of x ÷ coefficient of x 3

αβ + βγ + αγ = c/a

(0)(- 3) + (- 3) (5) + (0) (5) = – 15/1

= – 15 = – 15

Product of all the zeroes = – (constant term) ÷ coefficient of x 3

αβγ = – d/a

(0)(- 3)(5) = 0

5. 2 x 2 +(7/2) x +3/4

2 x 2 +(7/2) x +3/4

The equation can also be written as,

8x 2 +14x+3

8x 2 +12x+2x+3

4x (2x+3) +1(2x+3)

(4x+1)(2x+3)

4x+1=0 ⇒ x = -1/4

2x+3=0 ⇒ x = -3/2

Therefore, zeroes are -1/4 and -3/2

(- 3/2) + (- 1/4) = – (7)/4

= – 7/4 = – 7/4

(- 3/2)(- 1/4) = (3/4)/2

Exercise 2.4

1. For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorisation.

(i) (–8/3), 4/3

(ii) 21/8, 5/16

(iii) -2√3, -9

(iv) (-3/(2√5)), -½

(i) Sum of the zeroes = – 8/3

Product of the zeroes = 4/3

P(x) = x 2 – (sum of the zeroes) + (product of the zeroes)

Then, P(x)= x 2 – (-8x)/3 + 4/3

P(x)= 3x 2 + 8x + 4

Using splitting the middle term method,

3x 2 + 8x + 4 = 0

3x 2 + (6x + 2x) + 4 = 0

3x 2 + 6x + 2x + 4 = 0

3x(x + 2) + 2(x + 2) = 0

(x + 2)(3x + 2) = 0

⇒ x = -2, -2/3

(ii) Sum of the zeroes = 21/8

Product of the zeroes = 5/16

Then, P(x)= x 2 – 21x/8 + 5/16

P(x)= 16x 2 – 42x + 5

16x 2 – 42x + 5 = 0

16x 2 – (2x + 40x) + 5 = 0

16x 2 – 2x – 40x + 5 = 0

2x (8x – 1) – 5(8x – 1) = 0

(8x – 1)(2x – 5) = 0

⇒ x = 1/8, 5/2

(iii) Sum of the zeroes = – 2√3

Product of the zeroes = – 9

Then, P(x) = x 2 – (-2√3x) – 9

x 2 + 2√3x – 9 = 0

x 2 + (3√3x – √3x) – 9 = 0

x(x + 3√3) – √3(x + 3√3) = 0

(x – √3)(x + 3√3) = 0

⇒ x = √3, -3√3

(iv) Sum of the zeroes = -3/2√5x

Product of the zeroes = – ½

Then, P(x)= x 2 -(-3/2√5x) – ½

P(x)= 2√5x 2 + 3x – √5

2√5x 2 + 3x – √5 = 0

2√5x 2 + (5x – 2x) – √5 = 0

2√5x 2 – 5x + 2x – √5 = 0

√5x (2x + √5) – (2x + √5) = 0

(2x + √5)(√5x – 1) = 0

⇒ x = 1/√5, -√5/2

2. Given that the zeroes of the cubic polynomial x 3 – 6 x 2 + 3 x + 10 are of the form a , a + b , a + 2 b for some real numbers a and b , find the values of a and b as well as the zeroes of the given polynomial.

Given that a, a+b, a+2b are roots of given polynomial x³-6x²+3x+10

Sum of the roots ⇒ a+2b+a+a+b = -coefficient of x²/ coefficient of x³

⇒ 3a+3b = -(-6)/1 = 6

⇒ 3(a+b) = 6

⇒ a+b = 2 ——— (1) b = 2-a

Product of roots ⇒ (a+2b)(a+b)a = -constant/coefficient of x³

⇒ (a+b+b)(a+b)a = -10/1

Substituting the value of a+b=2 in it

⇒ (2+b)(2)a = -10

⇒ (2+b)2a = -10

⇒ (2+2-a)2a = -10

⇒ (4-a)2a = -10

⇒ 4a-a² = -5

⇒ a²-4a-5 = 0

⇒ a²-5a+a-5 = 0

⇒ (a-5)(a+1) = 0

a-5 = 0 or a+1 = 0

a = 5 a = -1

a = 5, -1 in (1) a+b = 2

When a = 5, 5+b=2 ⇒ b=-3

a = -1, -1+b=2 ⇒ b= 3

∴ If a=5 then b= -3

If a= -1 then b=3

3. Given that √ 2 is a zero of the cubic polynomial 6 x 3 + √ 2 x 2 – 10 x – 4 √ 2 , find its other two zeroes.

Given, √2 is one of the zero of the cubic polynomial.

Then, (x-√2) is one of the factor of the given polynomial p(x) = 6x³+√2x²-10x- 4√2.

So, by dividing p(x) by x-√2

NCERT Exemplar Solutions for Class 10 Maths Chapter 2-1

6x³+√2x²-10x-4√2= (x-√2) (6x² +7√2x + 4)

By splitting the middle term,

(x-√2) (6x² + 4√2x + 3√2x + 4)

= (x-√2) (2x+√2) (3x+2√2)

To get the zeroes of p(x),

Substitute p(x)= 0

(x-√2) (2x+√2) (3x+2√2)= 0

x= √2 , x= -√2/2 ,x= -2√2/3

Hence, the other two zeroes of p(x) are -√2/2 and -2√2/3

The exemplars for Chapter 2 Polynomials are provided in PDF, which can be downloaded easily. Students can use these solved questions as a reference tool while preparing for the board exam to clear their doubts. These exemplar problems and solutions have been designed by subject experts with respect to the CBSE Syllabus and as per the exercise questions available in the NCERT book . Click here to get exemplars for all chapters.

Students of Class 10 can also go through the online learning materials such as notes , exemplar books, and question papers available in downloadable PDFs to prepare and score well in the board exam. They are also advised to solve sample papers and previous years’ question papers to get an idea of the type of questions asked from the chapter Polynomials and the marking scheme for the same.

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NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.3

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NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4

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