maths assignment for class 10 polynomials

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

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.

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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.

• RS Aggarwal
• ML Aggarwal
• Merchant of Venice
• NCERT Books
• Questions and Answers
• NCERT Notes
• Important Questions
• Polynomials

NCERT Solutions for Chapter 2 Polynomials Class 10 Maths

The graphs of y = p (x) are given in the figure given below, for some polynomials p (x). Find the number of zeroes of p (x), in each case.

(i) x 2  - 2x - 8 We have p (x) = x 2  - 2x - 8 = x 2  + 2x - 4x - 8 = x (x + 2) - 4 (x + 2) = (x - 4) (x + 2) For p (x) = 0, we have (x - 4) (x + 2) = 0 Either x - 4 = 0 ⇒ x = 4 or x + 2 = 0 ⇒ x = - 2 ∴ The zeroes of x 2  - 2x - 8 are 4 and - 2.

Thus, relationship between zeroes and the coefficients in x 2  - 2x - 8 is verified.

(ii) 4s 2  - 4s + 1 We have p (s) = 4s 2  - 4s + 1 = 4s 2  - 2s - 2s + 1 = 2s (2s - 1) - 1 (2s - 1) = (2s - 1) (2s - 1) For p (s) = 0, we have, (2s - 1) = 0 ⇒ s = 1/2 ∴ The zeroes of 4s 2  - 4s + 1 are 1/2 and 1/2 Now,

Thus, the relationship between the zeroes and coefficients in the polynomial 4s 2  - 4s + 1 is verified.

(iii) 6x 2  - 3 - 7x We have p (x) = 6x 2  - 3 - 7x = 6x 2  - 7x - 3 = 6x 2  - 9x + 2x - 3 = 3x (2x - 3) + 1 (2x - 3) = (3x + 1) (2x - 3) For p (x) = 0, we have,

Either (3x+1) = 0

Thus, the relationship between the zeroes and the coefficients in the polynomial 6x 2  - 3 - 7x is verified.

(iv) 4u 2  + 8u We have, f(u) = 4u 2  + 8u = 4u (u + 2) For f (u) = 0, Either 4u = 0 ⇒ u = 0 or u + 2 = 0 ⇒ u = - 2 ∴ The zeroes of 4u 2  + 8u are 0 and - 2. Now, 4u 2  + 8u can be written as 4u 2  + 8u + 0.

Thus, the relationship between the zeroes and the coefficients in the polynomial 4u 2  + 8u is verified.

(v) t 2  - 15 We have, f (t) = t 2  - 15 = (t) 2  - (√15) 2 ( t+√15) = (t- √15) For f (t) = 0, we have Either ( t+√15) = 0 ⇒ t = -√15 or (t- √15) = 0 ⇒ t = √15 ∴ The zeroes of t 2  - 15 are -√15 and √15. Now, we can write t 2  - 15 as t 2  + 0t - 15.

Thus, the relationship between the zeroes and the coefficients in the polynomial t 2  - 15 is verified.

(vi) 3x 2  – x – 4 We have, f (x) = 3x 2  - x - 4 = 3x 2  + 3x - 4x - 4 = 3x (x + 1) - 4 (x + 1) = (x + 1) (3x - 4) For f (x) = 0 ⇒ (x + 1) (3x - 4) = 0 Either (x + 1) = 0 ⇒ x = - 1 or 3x - 4 = 0 ⇒ x = 4/3 ∴The zeroes of 3x 2  - x - 4 are 1 and Now,

Thus, the relationship between the zeroes and the coefficients in 3x 2  - x - 4 is verified.

Find a quadratic polynomial each with the given numbers as sum and product of its zeroes respectively:

(i) 1/4 , -1

Note: A quadratic polynomial whose zeroes are α and β is given by p (x) = {x 2  -(α + β) x + αβ} i.e., p (x) = {x 2  - (sum of the zeroes) x + (product of the zeroes)}

(i) Sum of the zeroes, (α + β) =  Product of the zeroes, ab = - 1 ∴ The required quadratic polynomial is x 2  - (α + β) x + ab

(ii) Sum of the zeroes, (α + β) = √2 Product of zeroes, αβ = 1/3  ∴The required quadratic polynomial is

(v) Since, sum of zeroes, (α+β) = -1/4  Product of zeroes, αβ = 1/4 ∴ The required quadratic polynomial

(i) Here, dividend p (x) = x 3  - 3x 2  + 5x - 3 divisor g (x) = x 2  - 2 ∴ We have

Thus, the quotient = (x - 3) and remainder = (7x - 9)

(ii) Here, dividend p (x) = x 4  - 3x 2  + 4x + 5 = x 4  + 0x 3  - 3x 2  + 4x + 5 and divisor g (x) = x 2  + 1 - x = x 2  - x + 1 ∴ We have

Thus, the quotient is (x 2  + x - 3) and remainder = 8

(iii) Here, dividend, p (x) = x 4  - 5x + 6 = x 4  + 0x 3  +0x 2 - 5x + 6 and divisor, g (x) = 2 - x 2  = - x 2  + 2 ∴ We have

Thus, the quotient = –x 2  – 2 and remainder = –5x + 10

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

(i)  t 2  – 3,  2 t 4  + 3 t 3  – 2 t 2  – 9 t  – 12

(ii)  x 2  + 3 x + 1,  3 x 4  + 5 x 3  – 7 x 2  + 2 x  + 2

(iii)  x 3  – 3 x + 1,  x 5  – 4 x 3  +  x 2  + 3 x  + 1

(i) Dividing 2t 4  + 3t 3  - 2t 2  - 9t - 12 by t 2  - 3, we have:

∵ Remainder = 0 ∴ (t 2  - 3) is a factor of 2t 4  + 3t 3  - 2t 2  - 9t - 12.

(ii) Dividing 3x 4  + 5x 3  - 7x 2  + 2x + 2 by x 2  + 3x + 1, we have:

∵ Remainder = 0. ∴ x 2  + 3x + 1 is a factor of 3x 4  + 5x 3  - 7x 2  + 2x + 2.

(iii) Dividing x 5  - 4x 3  + x 2  + 3x + 1 by x 3  - 3x + 1, we get:

∴ The remainder = 2, i.e., remainder ≠ 0 ∴ x 3  - 3x + 1  is not a factor  of x 5  - 4x 3  + x 2  + 3x + 1.

We have p (x) = 3x 4  + 6x 3  - 2x 2  - 10x - 5

Here, Dividend p (x) = x 3  - 3x 2  + x + 2 Divisor = g (x) Quotient = (x - 2) Remainder = (-2x + 4) Since, (Quotient × Divisor) + Remainder = Dividend

∴ [(x − 2) × g (x)] + [(−2x + 4)] = x 3  − 3x 2  + x + 2 ⇒ (x − 2) × g (x)= x 3  − 3x 2  + x + 2 − (−2x + 4) = x 3  − 3x 2  + x + 2 + 2x −4

= x 3  - 3x 2  + 3x - 2

We have (i)   p (x) = 3x 2  - 6x + 27 g (x) = 3 q (x) = x 2  - 2x + 9 r (x) = 0

p (x)= q (x) × g (x) + r (x).

(ii) p (x)= 2x 3  − 2x 2  + 2x + 3 g (x) = 2x 2  − 1 q (x) = x − 1 r (x) = 3x + 2

⇒ p (x)= q (x) × g (x) + r (x)

(iii) p (x) = 2x 3  − 4x 2  + x + 4 g (x) = 2x 2  + 1 q (x)= x − 2 r (x) = 6

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

(i) 2x 3  + x 2 - 5x + 2;   1/2, 1 , -2 

Let p (x) = 2x 3  + x 2  - 5x + 2

Comparing with  a x 3  + bx 2  +cx+d

a  = 2, b = 1, c = -5, d = 2

Again, p (1) = 2 (1) 3  + (1) 2  - 5 (1) + 2 = 2 + 1 - 5 + 2 = (2 + 2 + 1) - 5 = 5 - 5 = 0 ⇒ 1 is a zero of p (x). Also p (- 2) = 2 (- 2) 3  + (- 2) 2  - 5 (- 2) + 2 = 2(- 8) + (4) + 10 + 2 = - 16 + 4 + 10 + 2 = - 16 + 16 = 0 ⇒ -2 is a zero of p (x). Relationship ∵ p (x) = 2x 3  + x 2  - 5x + 2 ∴ Comparing it with ax 3  + bx 2  + cx + d, we have : a = 2, b = 1, c = - 5 and d = 2 Also 1/2 , 1 and - 2 are the zeroes of p (x) Let α = 1/2 , β = 1 and γ = - 2

Thus, the relationship between the co-efficients and the zeroes of p (x) is verified.

(ii) Here, p (x) = x 3  - 4x 2  + 5x - 2 ∴ p (2) = (2) 3  - 4 (2) 2  + 5 (2) - 2 = 8 - 16 + 10 - 2 = 18 - 18 = 0 ⇒ 2 is a zero of p (x) Again p (1) = (1) 3  - 4 (1) 2  + 5 (1) - 2 = 1 - 4 + 5 - 2 = 6 - 6 = 0 ⇒ 1 is a zero of p (x). ∴ 2, 1, 1 are zeroes of p (x). Now, Comparing p (x) = x 3  - 4x 2  + 5x - 2 with ax 3  + bx 2  + cx + d = 0, we have a = 1, b = -4, c = 5 and d = -2 ∵ 2, 1, and 1 are the zeroes of p (x) ∴ Let α = 2 β = 1 γ = 1 Relationship, α + β + γ = 2 + 1 + 1 = 4

Thus, the relationship between the zeroes and the co-efficients of p (x) is verified.

Let the required cubic polynomial be ax 3  + bx 2  + cx + d and its zeroes be a, b and g.

If a = 1, then -b/a = 2 ⇒ b = - 2 c/a = - 7 ⇒ c = - 7 -d/a = - 14 ⇒ d = 14 ∴ The required cubic polynomial = 1x 3  + (-2)x 2  + (-7) x + 14 = x 3  - 2x 2  - 7x + 14

We have p (x) = x 3  - 3x 2  + x + 1 Comparing it with ax 3  + bx 2  + cx + d. We have a = 1, b = - 3, c = 1 and d = 1 ∵ It is given that (a - b), a and (a + b) are the zeroes of the polynomial. ∴ Let, α = (a - b) β = a and γ = (a + b)

Again, αβγ = -d/a = -1 ⇒ (a - b) × a × (a + b) = - 1 ⇒ (1 - b) × 1 × (1 + b) = - 1  [∵ a = 1, proved above] ⇒ 1 - b 2  = - 1 ⇒ b 2  = 1 + 1 = 2 ⇒ b = ± √2 Thus, a = 1 and b = ±√2

Here, p (x) = x 4  - 6x 3  - 26x 2  + 138x - 35. ∵ Two of the zeroes of p (x) are : 2± 

∴[x-(2+√3)] [x-(2-√3)]

or, [(x-2)-√3] [(x-2)+√3] or, (x - 2) 2  - (√3) 2 or, (x 2  + 4 - 4x) - 3 or, x 2  - 4x + 1 is a factor of p (x). Now, dividing p (x)  by  x 2  - 4x + 1 we have,

Given that if the polynomial, x 4  - 6x 3  + 16x 2  - 25x + 10 is divided by another polynomial, x 2  - 2x + k, then remainder comes out is (x+a). So, we divide x 4  - 6x 3  + 16x 2  - 25x + 10 by x 2  - 2x + k to find quotient and remainder.

∵ Remainder = (2k - 9)x - k (8 - k) + 10 But the remainder = x + α

Applying the division algorithm to the polynomials x 4  - 6x 3  + 16x 2  - 25x + 10 and comparing the like coefficients,

Thus, k = 5 and α = -5

NCERT Solutions for Chapter 4 The Age of Industrialisation Class 10 History

Related chapters.

• Real Numbers
• Pair of Linear Equations in Two Variables
• Quadratic Equations
• Arithmetic Progressions

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• NCERT Revision Notes for Chapter 2 Polynomials Class 10 Maths

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Class 10 Mathematics Polynomials Assignments

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

Mathematics Polynomials Class 10 Assignments Pdf Download

We have provided below the biggest collection of free CBSE NCERT KVS Assignments for Class 10 Mathematics Polynomials . Students and teachers can download and save all free Mathematics Polynomials assignments in Pdf for grade 10th. Our expert faculty have covered Class 10 important questions and answers for Mathematics Polynomials as per the latest syllabus for the current academic year. All test papers and question banks for Class 10 Mathematics Polynomials and CBSE Assignments for Mathematics Polynomials 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 Polynomials Download in Pdf

Advantages of Class 10 Mathematics Polynomials Assignments

• As we have the best and largest collection of Mathematics Polynomials 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 Polynomials textbooks for Class 10 .
• All Mathematics Polynomials 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 Polynomials chapter wise worksheets and assignments for free in Pdf
• Class 10 Mathematics Polynomials question bank will help to improve subject understanding which will help to get better rank in exams

Frequently Asked Questions by Class 10 Mathematics Polynomials students

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

We provide here Standard 10 Mathematics Polynomials 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 Polynomials 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 Polynomials 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 for class 10 maths Chapter 2 Polynomials will help the students revisit concepts that they might have studied till now in their previous classes. Polynomial is an expression that consists of variables, coefficients, with various mathematical operations like addition , subtraction , multiplication , division, and a non-negative integer exponent. Also, the highest power of x or the variable in a polynomial is known as the degree of that polynomial. A polynomial can be further classified on this basis into linear, quadratic, cubic polynomials. Kids are given a quick revision of concepts such as the categorization of polynomials as well as the zeros of a polynomial before proceeding to seamlessly integrate new topics into their curriculum. The NCERT solutions class 10 maths Chapter 2 Polynomials kids will be introduced to the geometrical meaning of zeros of a polynomial, the relationship between zeros and coefficients of a polynomial, as well as the division algorithm for polynomials.

All of the topics covered in this chapter will help students in higher grades, as the above-mentioned topics form the basis of algebra. Lastly, the division algorithm for polynomials is also illustrated by well-drafted examples which the students must take note of. The class 10 maths NCERT Solutions Chapter 2 Polynomials with all the facts and formulas regarding polynomials are given below and also you can find some of these in the exercises given below.

• NCERT Solutions Class 10 Maths Chapter 2 Ex 2.1
• NCERT Solutions Class 10 Maths Chapter 2 Ex 2.2
• NCERT Solutions Class 10 Maths Chapter 2 Ex 2.3
• NCERT Solutions Class 10 Maths Chapter 2 Ex 2.4

NCERT Solutions for Class 10 Maths Chapter 2 PDF

These NCERT solutions in class 10 maths will give enough practice to the students on how to find the zeroes of the given quadratic polynomials, and how to verify the relationship between the zeroes and the coefficients. Also, they will get to learn how to proceed with the division of polynomials thereby finding their quotients and remainders . The strategic approach towards all of the above topics is neatly outlined in each of the sections of the NCERT Solutions Class 10 Maths below:

☛ Download Class 10 Maths NCERT Solutions Chapter 2 Polynomials

NCERT Class 10 Maths Chapter 2   Download PDF

The concepts mentioned above of polynomials serve as the foundation for advanced mathematics , therefore mastering the fundamentals by completing a variety of problems is important to instill a basic understanding of the topic. The solved examples presented in the book will help the students grasp the polynomial properties effectively. An exercise-wise analysis of the NCERT Solutions Class 10 Maths Chapter 2 Polynomials is provided below:

• Class 10 Maths Chapter 2 Ex 2.1 - 1 Question
• Class 10 Maths Chapter 2 Ex 2.2 - 2 Questions
• Class 10 Maths Chapter 2 Ex 2.3 - 5 Questions
• Class 10 Maths Chapter 2 Ex 2.4 - 5 Questions

Topics covered : The topics covered in the class 10 maths NCERT Solutions Chapter 2 are degrees of polynomials , quadratic polynomial , cubic polynomials , the relationship between zeroes, and the coefficients of the given polynomials, and the division algorithm of polynomials.

Total Questions: Class 10 Maths Chapter 2 Polynomials consist of 13 key questions, of which 10 are easy, 3 are medium-level sums and 6 problems that require some brainstorming as they are challenging.

List of Formulas in NCERT Solutions Class 10 Maths Chapter 2

NCERT Class 10 maths chapter 2 encompass different concepts of polynomials. The formulas are given in this lesson work to simplify tough problems that would otherwise prove to be very time-consuming. Kids are required to learn certain formulas and processes that are crucial to attempting the questions on polynomial.s This is a topic that will make its way into the chapters of higher-order studies directly or indirectly. Thus, building a deep-seated knowledge base for the same is very important. Some of the important theories and formulas covered in the NCERT Solutions Class 10 Maths Chapter 2 are given below :

• Degree of a polynomial: p(x), where x is the highest power of the polynomial
• The graph of a linear polynomial is given as ax + b which is a straight line
• Relationship between zero and coefficient i.e k= - constant term/ coefficient of x (linear polynomial).
• For a quadratic polynomial:

Sum of its zeros = - (coefficient of x)/(coefficient of x 2 )

Product of its zeros = (constant term)/(coefficient of x 2 )

• According to the division algorithm, for any polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x)

Important Questions for Class 10 Maths NCERT Solutions Chapter 2

Video solutions for class 10 maths ncert chapter 2, faqs on ncert solutions class 10 maths chapter 2, why are ncert solutions class 10 maths chapter 2 vital for scoring well.

NCERT Solutions Class 10 maths chapter 2 Polynomials have surprising applications not just in mathematics but in everyday life as well. Since polynomials are used to define curves, they are used in online banking, high-frequency trading algorithms, computers, aircraft, and many more areas. The concept of polynomials is applied in higher-level mathematics at a large scale. Hence, it is important to have a strong foundation of this topic to score well in exams.

Do I Need to Practice all Questions Given in NCERT Solutions Class 10 Maths Polynomials?

The questions for the board examinations are mainly picked from NCERT Solutions. Questions on linear, quadratic, and cubic polynomials, the meaning of zero of a polynomial, and real numbers need regular practice. The CBSE also recommends studying from the NCERT Solutions Class 10 Maths Polynomials solutions, thereby making them very important. Hence, it is advisable for the students to solve all the questions to gain enough practice.

What are the Important Topics Covered in NCERT Solutions Class 10 Maths Chapter 2?

The important topics covered in NCERT solutions class 10 maths chapter 2 are the definition of a zero of a polynomial, degrees of a polynomial, and the division algorithm of polynomials. Polynomials Chapter 2 is very diverse and helps the students understand concepts such as parabolas, graphs, and the geometric meaning of zeros of the polynomial.

How Many Questions are there in NCERT Solutions Class 10 Maths Chapter 2 Polynomials?

The NCERT Solutions class 10 maths chapter 2 Polynomials consist of 13 well-researched questions. These 13 sums are well-placed concerning the theory's setting. These problems are in different formats such as equation, long answer type, word problems, proof-based questions, and many more. The variety of questions make the NCERT solutions interesting and engaging.

What are the Important Formulas in NCERT Solutions Class 10 Maths Chapter 2?

The important formulas and concepts covered in the NCERT Solutions class 10 chapter 2 include the division algorithm, the relationship between zeroes and the coefficient, linear, quadratic, and cubic polynomial equations. The formulas can help students solve questions faster and complete the exam syllabus well within the given time frame.

Why Should I Practice NCERT Solutions Class 10 Maths Polynomials Chapter 2?

NCERT Solutions assists the students in reinforcing their understanding of concepts like coefficients, parabolas, variables, and many more. NCERT Solutions class 10 maths Polynomials chapter 2 include graphs and geometry. While attempting questions from this chapter that are based on real-life situations, students will develop problem-solving skills. This is one of the most important chapters for board exams; hence students must practice all the questions.

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NCERT Solutions For Class 10 Maths Chapter 2 Polynomials: Download PDF

Sushmita Sundas ,

Nov 18, 2023

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The NCERT solutions for class 10 Maths chapter 2 Polynomials include topics such as Geometrical Meaning of the Zeros of Polynomial, Division Algorithm for Polynomials and more. Students can download the detailed PDF here.

NCERT solutions for class 10 Maths chapter 2 Polynomials are provided here to assist students in preparing well for their class 10th board exams. These solutions have been prepared by subject specialists. These subject specialists have prepared these solutions in a way that makes it simpler for students to solve the Chapter 2 Polynomials questions. The addition of step-by-step explanations to these Maths NCERT Class 10 solutions makes learning easier for students.

A polynomial expression that consists of variables, constants, and exponents and that is combined using mathematical operations such as addition, subtraction, multiplication, and division. Additionally, when creating these solutions, the subject experts followed the revised CBSE 10th syllabus for 2023–24 . Given below are the NCERT Solutions Class 10 Maths Chapter 2.

Table of Contents :

• NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Download PDF

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Important Questions

Ncert solutions for class 10 maths chapter 2 topics & sub-topics, ncert solutions for class 10 maths chapter 2 important features, ncert solutions for class 10 maths chapter 2 polynomials download pdf.

NCERT Solutions for Class 10 Maths Chapter 2 can be downloaded in the form of a pdf file. Polynomials class 10 ncert solutions are provided in the table below for students to download:

The solutions to a variety of questions on polynomials and their applications are covered in Class 10 Maths chapter 2 solutions. Students will get to study about the zeroes of quadratic polynomials connected to their coefficients as well as the division procedure for polynomials of integers. Given below are the important questions of  NCERT Solutions for Class 10 Maths Chapter 2:

This is one of the most significant topics, as it falls under the Algebra unit, which is worth 20 marks in the Class 10 Maths CBSE examination. There is generally 1 question from this chapter in the board exam. Given below are the topics & subtopics of NCERT solutions for class 10 Maths chapter 2 Polynomials:

The unit Algebra in the CBSE class 10 syllabus 2023–24 includes the chapter about Polynomials. To ensure that the students fully grasp the idea of Polynomials, the NCERT solutions are provided. The following are the features of NCERT solutions for class 10 Maths chapter 2 Polynomials:

• It follows the Class 10 Mathematics CBSE Syllabus for 2023–24.
• Students can feel certain that they will perform well in the exams after studying these NCERT solutions created by subject specialists.
• It adheres to NCERT norms, which aid in the student’s appropriate preparation.
• From the exam point of view, it comprises all the crucial questions and their solutions.

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Maths assignment class viii | quadrilateral, lesson plan math class x (ch-2) | polynomials.

• Basic concepts and definitions related to the topic polynomials.
• Explanation of remainder and factor theorem.
• Relationships between zeroes and coefficients of quadratic polynomials and cubic polynomials.
• Explanation of division algorithm.
• Method of finding all zeroes of cubic and bi-quadratic polynomials.
• After studying this lesson students will be able    to explain the relationship between zeroes and coefficients. 
• Students should know the method of division of one polynomial with another and the division algorithm. 
• Students should be able to factorize the quadratic, cubic and bi-quadratic polynomials and become able to find their zeroes.

Introduction and motivation

Teacher will start the session by giving many examples of Algebraic Expressions. Now teacher will explain which types of Algebraic Expressions are called Polynomials. After this teacher will introduce the definition of Polynomial and explain the different components of polynomials.

Polynomial: An algebraic expression in which the exponent of the variable is a whole number is called a polynomial.

Example: 3x 2  + 5,  4x 3  - 8x 2  + 5x - 3,  5x 4  + 3x 2    are some examples of polynomial .

Important Definitions

Now teacher will explain the Important Definitions like Degree, coefficients and zeroes of the polynomials and their graphical representations.

Degree of constant polynomial is zero and degree of zero polynomial is undefined.

After this teacher will explain the different types of polynomial on the bases of degree and on the basis of number of terms. 

 Teacher will also give examples  and explanations of definitions of the polynomials like linear, quadratic, cubic, bi-quadratic, monomial, binomial, and tri-nomial .

Finding Zeroes from the Graph

Students will be given the knowledge of finding the number of zeroes from the graph of the polynomial.

Number of zeroes of a polynomial is equal to the number of times the graph of that polynomial meet the x - axis.

Graph of a quadratic polynomial is always parabolic.

Zeroes of Quadratic Polynomial

Now teacher will explain the Factor method of finding the zeroes of quadratic polynomials and will explain the Relationship between zeroes and coefficients. 

Teacher will also provide some problems to the students to check whether students can apply the method or not.

Now teacher will explain the method of finding the quadratic polynomial from the zeroes and explain the topic by taking some examples.

Division Algorithm

By taking some examples, teacher will explain the method of dividing one polynomial by another. To verify the result of division teacher will explain the Division Algorithm.

Dividend  = Divisor  ❌  Quotient + Remainder

Zeroes of Bi-Quadratic Polynomial

Now teacher will explain the method of finding all zeroes of bi quadratic polynomials by using factor theorem and division algorithm. Method will be well explained by taking examples. 

Teacher can also give some problems to the students to check whether they will understand the concept or not.

Zeroes of Cubic Polynomial

Now concept of cubic polynomial is also be introduced. Students will be given the knowledge of cubic polynomial, its general form, its zeroes and the relationship between zeroes and  coefficient.

• Review questions given by the teacher. 
• Student can prepare the presentation on the topic relationship between zeroes and coefficients for quadratic as well as cubic polynomials. 
• Solve NCERT problems with examples, 
• Solve assignment on Multiple Choice Questions (MCQ) given by the teacher.

Students can extend their learning through the RESOURCE CENTRE and can find more valuable and interesting concepts on mathematics at  cbsemathematics.com

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Lesson Plan Math Class 10 (Ch-1) | Real Numbers

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Course: Class 10   >   Unit 2

• Quiz 4 Polynomials