# NCERT Solutions for Class 10 Maths Chapter 2 Polynomials (Ex 2.2) Exercise 2.2

Mathematics is one of the most important subjects that are taught in Class 10. It is one of the subjects that has high utility even after school. The use of Mathematics does not end with its academic significance. Mathematics helps lay the groundwork for a number of fields. Mathematics is required for many career fields, including flying, engineering, financial planning, being a mathematician, an economist, a data scientist, and a market researcher.It is important to have a basic knowledge of Mathematics not just to enhance career opportunities but also to conduct general tasks like managing finances, understanding income tax, and many more. Mathematics is a subject that has high utility in the everyday lives of almost everyone. Students should have a good grasp of the mathematical concepts. Having a good understanding of mathematical concepts is essential for competitive examinations like the CAT, AFCAT, etc., since they help develop logical reasoning skills.

Students should consult the NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2.The CBSE board is in charge of conducting annual board examinations.It is highly influenced by the NCERT curriculum. The NCERT curriculum is prepared from the CBSE board examination perspective, hence, it is advised that students thoroughly study the NCERT exercise solutions that can be accessed from the Extramarks website. Students may also resort to the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 for a thorough understanding of the concepts covered in Polynomials. It is recommended that students practise the NCERT Solutions to be able to score better in the CBSE board examination of Class 10.

The Central Board of Secondary Education is in charge of annually conducting board examinations for the students of secondary schools. Students of Class 10 can feel apprehensive about the board examination, since they appear for it for the first time in Class 10. Students are advised to go through the NCERT exercise problems and solutions provided on the Extramarks website. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can be referred to by students from the Extramarks website. It is suggested that students consistently practise the NCERT exercise to gain confidence in relation to the CBSE board examination. Students must understand the importance of securing high marks in the CBSE Board Examination of Class 10. Based on the grades secured by students in Class 10, students get to select the field of their choice in the senior secondary classes. Students are required to understand that practise is key to scoring higher marks in the CBSE Board Examination of Class 10. Students can access study material to study for the CBSE board examination from the Extramarks website. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can be downloaded from the Extramarks website in PDF format.

**NCERT Solutions for Class 10 Maths Chapter 2 Polynomials (Ex 2.2) Exercise 2.2**

Polynomials is one of the most important topics that is covered in the Mathematics syllabus of Class 10. The compilation of variables, constants and exponents that makes up a mathematical expression is called a Polynomial. Polynomials are an important part of the mathematical language. Polynomials also play a vital role in algebra.Class 10 Maths Chapter Exercise 2.2 is one of the most difficult and important exercises in the polynomials chapter.It carries a lot of weight in the CBSE Board Examination for Class 10.Students should practise the Class 10 Maths Chapter 2 Exercise 2.2 Solutions to work on their basic mathematical concepts. Mathematics can be a complex subject to understand, it requires constant practise from the students’ end to have a better understanding of the subject. To have a good command of the subject, students must put in consistent effort to understand the fundamental mathematical concepts.Maths Class 10 Chapter 2 Exercise 2.2 holds a lot of weightages in the CBSE board examination. Students must study and practise the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 to score well in the CBSE board examination. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can also be accessed by students in a downloadable PDF format.

To understand Class 10 Mathematics in particular, the student must make a conscious effort to understand the mathematical concepts.This is crucial to the Mathematics results of students in the CBSE board examination. Students have the option to download the required study material in PDF format from Extramarks. This feature has been provided by Extramarks to make it easier and more effortless for students to access the study material they need. A PDF format of the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 is also available on the Extramarks website for students to download. Students can download the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 PDF file to practise and prepare for the Class 10 CBSE Board Examination of Mathematics. Often, students find themselves unable to access online study material due to many reasons, like internet connection issues.Keeping this in mind, Extramarks has equipped students with the option to download any study material necessary, in PDF format, from the Extramarks website. Students of Class 10 appearing for the CBSE Board Examination of Mathematics can access the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 to refer to for their preparation. Along with PDF files of exercise problems solutions, students can also download PDF files of other study material like sample papers, revision notes, past years’ papers, etc. Downloadable PDF files of all subjects in all chapters are available on the Extramarks website for students to access and download. Students are advised to download the PDF file of the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 to have easier access.

**Access NCERT Solutions for Class 10 Mathematics Chapter 2 – Polynomials**

Extramarks is an educational website. It provides services to students with the objective of making learning resources available to them and enhancing their learning experience. Extramarks also has a mobile application that students can download on their mobile phones to access study material more effortlessly. Students of Class 10 can refer to study material available on the Extramarks website to enrich their knowledge and enhance their preparation process. Mathematics is a tough subject to grasp. Class 10 students must be diligent about the CBSE Board Examination of Mathematics. Students in Class 10 should refer to the NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2, to polish their mathematical concepts.Mathematics is a significant and demanding subject.It takes hard work and consistent practise to master this subject. To be able to secure higher marks in the CBSE Board Examination of Mathematics, students should understand the foundational concepts of Mathematics in detail. To improve their grades, students must regularly practise the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2It is suggested that students stay consistent with their practice. Students are encouraged to start preparing for the CBSE board examination from the very beginning of the academic year. Students who start preparing for the CBSE board examination right from the beginning find it relatively easier than most. Students must go through Mathematics syllabus carefully and thoroughly. Mathematics is a technical subject. To master it, it is essential for students to have a good sense of mathematical concepts. It is important that students have a clear understanding of the basic concepts of Mathematics to secure high academic results in the CBSE board examination. Students are advised to use the NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2, in order to achieve higher marks in the CBSE Class 10 Board Examination.

Students of all classes can have access to the study material provided by Extramarks. Extramarks caters to the learning requirements of all students from Class 1 to Class 12.

Students of Class 1 can access the NCERT Solutions Class 1 from the Extramarks website. The NCERT Solutions Class 2 can be found on the Extramarks website for the students to refer to. Students may access the NCERT Solutions Class 3 of all subjects in PDF format.

Extramarks. Students of Class 4 are suggested to refer to the NCERT Solutions Class 4 on Extramarks for solutions to all exercise problems. Students can download the NCERT Solutions Class 5 in PDF format from the Extramarks website or mobile application. The NCERT Solutions Class 6 can be availed by students from Extramarks. Students can find the NCERT Solutions Class 7 of all chapters of all subjects in an organised manner on the Extramarks website. The NCERT Solutions Class 8 are available on the Extramarks website for students of Class 8. Students of Class 9 may refer to the NCERT Solutions Class 9 for better preparation. Students can practise the NCERT Solutions Class 10 to be better prepared for the CBSE board examination. The NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2, can also be used to prepare for the CBSE Board Examination in Mathematics.The NCERT Solutions Class 11 can be utilised by students of Class 11 for their preparation. It is recommended that students of Class 12 thoroughly go through the NCERT Solutions Class 12 of all subjects to have a better understanding of the concepts covered in the concerned syllabus. Importance of Polynomials

Polynomials are a very significant part of the mathematical language. Polynomials are important in almost all fields of Mathematics. They are used for expression in the mathematical language. They can be quite useful in algebra as well. General use of Polynomials has great importance in day to day life as well. Polynomials can be quite useful in everyday life. They play a vital role in studying and understanding traffic patterns. Polynomials are also used by Economists to model the economic growth pattern.Polynomials are also used in the medical research industry by researchers to describe the behaviour of bacterial colonies. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can prove useful for the preparation of students in Class 10.NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2

There are many websites online where students may acquire study material. It can be tough for students to obtain adequate resources for studying. The study material provided by Extramarks is regularly reviewed and updated. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 have been curated in a way that makes it convenient for students to use as a primary source of information for preparation. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 may help students feel more confident in relation to the CBSE board examination. Class 10 can be academically challenging for students, however, it is just as important as it is difficult. Extramarks provides resources to help students in the preparation of their examinations. Extramarks is one of the most renowned educational websites. It is trusted by students and parents, alike, for adequate study material. Students can access the resources provided by Extramarks by registering themselves on the Extramarks website. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can serve as the primary document to study from, students can refer to the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 directly for a thorough understanding of the NCERT Mathematics syllabus for Class 10 board examination. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 is accessible on the Extramarks website as well as the Extramarks mobile application. Students can also download the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 in PDF format.

**Definition **

A combination of algebraic terms consisting of a constant, multiplied by one or more non-negative variables, makes up a mathematical expression. This expression is called a Polynomial. A Polynomial always consists of more than two algebraic terms. A Polynomial is the compilation of constants, exponents, and variables. The chapter Polynomials is one of the most important chapters of the NCERT Class 10 Mathematics curriculum. It is a highly significant concept in Mathematics. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can help students understand the concept of Polynomials in detail. Students can prepare for this chapter thoroughly with the help of the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2.

**Degree of a Polynomial**

In a polynomial equation, the greatest or highest power denoted to a variable is said to be the degree of that polynomial equation. It is essential that students understand what the degree of a polynomial means. It is a concept that appears repeatedly in the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2, hence, it is important for students to have clarity about the concept of degree of a polynomial. Students may refer to the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 better prepare the chapter Polynomial. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can be accessed by students from the Extramarks mobile application or website.

**Types of Polynomial**

There are three types of polynomials that are covered in the NCERT Mathematics syllabus of Class 10, linear polynomial, quadratic polynomial, cubic polynomial. When the coefficients of the variables in a polynomial equation are zero, it is called a zero polynomial. A linear polynomial is a polynomial whose degree is one. A quadratic polynomial is a polynomial whose degree is 2. A cubic polynomial is a polynomial whose degree is 3. Students must make efforts to understand the type of polynomial thoroughly. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can be helpful in understanding the different types of polynomials.

**Value of a Polynomial**

The value of a polynomial is one of the most important concepts covered in the Class 10 Mathematics syllabus. Understanding the meaning of “value” in a polynomial is crucial to understanding the overall concept of Polynomials. It is important that students understand the basic concepts of the chapter Polynomials to effectively solve the exercise questions. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 must be utilised by students to solve the NCERT exercise questions. Students can also access additional learning resources from the Extramarks website to help them prepare for the CBSE board exam.

**Zero of Polynomial **

When the value of a polynomial as a whole becomes zero, it is defined as the zero of a polynomial. It is imperative that students understand the importance of understanding the meaning of zero in a polynomial. To score higher marks in the CBSE Board Examination of Mathematics of Class 10, students should go through the syllabus in detail. Students must thoroughly understand the fundamental concepts of mathematics.Students are suggested to practise the NCERT exercise problems and solutions regularly, since the CBSE board examination is widely based on the NCERT curriculum. Students may resort to the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 for the preparation of the CBSE board examination.

**Graph of a Polynomial**

The graph of a polynomial is a significant topic in the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2. Graph of a polynomial has a high topic weightage. Graph related questions tend to have a lot of weightages in the CBSE board examination. Students must make sure to practise graph questions repeatedly to secure higher marks in the CBSE Board Examination of Mathematics. The NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2, can be used to practise the relevant topic and gain a better understanding of it.Students are advised to utilise the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 for more detailed insight into the solutions to the NCERT exercise problems. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 have been prepared in an elaborate manner.

**Meaning of the Zeroes of a Quadratic Polynomial**

Zeros of a quadratic polynomial are a highly crucial part of Mathematics. This concept is heavily weighted in the CBSE board examination for mathematics class ten.Students must prepare for the CBSE board examination with utmost sincerity. The NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2, can be found on the Extramarks website and can be used to prepare for the CBSE board examination.

**The Following Observations Can Be Made**

Students may take note of the fact that the syllabus of the CBSE board examination is significantly based on the NCERT curriculum. The content of NCERT books incorporates the syllabus requirements of the CBSE board examination. Students who prepare using the NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2 have a better chance of passing the CBSE board exam.It is advisable for students to regularly practise the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 provided by Extramarks, which can be obtained from the Extramarks website or mobile application. Solved Examples

Extramarks provides study resources for students. The study material provided by Extramarks is adequate for the preparation of the CBSE board examination. Students can refer to resources available on the Extramarks website for their preparation. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can also be accessed by students for their preparation. The various resources that can be availed by students from the Extramarks website are sample papers, NCERT exercise solutions, past years’ papers, revision notes, solved examples. Students are encouraged to practise solved examples for a clear and firm understanding of the subject from an examination perspective. Practising solved examples can help students work on their speed, which can help them attempt the paper faster at the time of the examination.

**NCERT Solutions Class 10 Maths All Chapters**

It is important for students to understand the significance of the NCERT curriculum in the CBSE board examination. The NCERT book is the primary text of reference for the CBSE board examination. The class 10 mathematics syllabus is not just important for the CBSE board examination, but also for many other competitive exams. Students who intend to take competitive examinations should prioritise practising the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 on a regular basis.Students ought to thoroughly go through the NCERT Mathematics syllabus in its entirety. The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can be utilised by students for the same. Students must go through each chapter in-depth and properly understand the mathematical concepts. It is critical that students prioritise understanding and practising mathematical concepts because they serve as the foundation for overall skill development in the field of mathematics.To reinforce basic mathematical concepts, refer to the NCERT Solutions for Class 10 Maths, Chapter 2, Exercise 2.2.NCERT Solutions Class 10 Maths Chapter 2 Exercises

Students must constantly practise the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2.Practising the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can help students cover Class 10 Mathematics chapter 2 thoroughly. It is essential that students practise the NCERT exercise problems and solutions to better prepare for the CBSE board examination, as the CBSE board examination is highly based on the NCERT curriculum.

**Q.1 **

$\begin{array}{l}\text{Find the zeroes of the following quadratic polynomials}\\ \text{and verify the relationship between the zeroes and}\\ \text{the coefficients.}\\ \text{(i)}{\mathrm{x}}^{2}-2\mathrm{x}-8\text{(ii)}4{\mathrm{s}}^{2}-4\mathrm{s}+1\text{(iii)}6{\mathrm{x}}^{2}-3-7\mathrm{x}\\ \text{(iv)}4{\mathrm{u}}^{2}+8\mathrm{u}\text{(v)}{\mathrm{t}}^{2}-15\text{(vi)}3{\mathrm{x}}^{2}-\mathrm{x}-4\end{array}$

**Ans.**

$\begin{array}{l}\text{(i)}\\ \text{}{\mathrm{x}}^{2}-2\mathrm{x}-8={\mathrm{x}}^{2}-4\mathrm{x}+2\mathrm{x}-8\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\mathrm{x}(\mathrm{x}-4)+2(\mathrm{x}-4)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=(\mathrm{x}+2)(\mathrm{x}-4)\\ \text{So the value of}{\mathrm{x}}^{2}-2\mathrm{x}-8\text{is zero when}\mathrm{x}=-2\text{or}\mathrm{x}=4.\\ \text{Therefore, the zeroes of}{\mathrm{x}}^{2}-2\mathrm{x}-8\text{are}-2\text{and 4.}\\ \text{Now,}\\ \text{Sum of zeroes =}-2+4=2=\frac{-(-2)}{1}=\frac{-(\text{Coefficient of x)}}{{\text{Coefficient of x}}^{2}}\\ \text{Product of zeroes =}-2\times 4=\frac{-8}{1}=\frac{\text{Constant term}}{{\text{Coefficient of s}}^{2}}\\ \text{(ii)}\\ \text{}4{\mathrm{s}}^{2}-4\mathrm{s}+1={\left(2\mathrm{s}-1\right)}^{2}\\ \text{So the value of}4{\mathrm{s}}^{2}-4\mathrm{s}+1\text{is zero when}\mathrm{x}=\frac{1}{2}.\\ \text{Therefore, the zeroes of}4{\mathrm{s}}^{2}-4\mathrm{s}+1\text{are}\frac{1}{2}\text{and}\frac{1}{2}\text{.}\\ \text{Now,}\\ \text{Sum of zeroes =}\frac{1}{2}+\frac{1}{2}=1=\frac{-(-4)}{4}=\frac{-(\text{Coefficient of s)}}{{\text{Coefficient of s}}^{2}}\\ \text{Product of zeroes =}\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}=\frac{\text{Constant term}}{{\text{Coefficient of s}}^{2}}\\ \text{(iii)}\\ 6{\mathrm{x}}^{2}-3-7\mathrm{x}=6{\mathrm{x}}^{2}-9\mathrm{x}+2\mathrm{x}-3\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=3\mathrm{x}(2\mathrm{x}-3)+1(2\mathrm{x}-3)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=(3\mathrm{x}+1)(2\mathrm{x}-3)\\ \text{So the value of}6{\mathrm{x}}^{2}-3-7\mathrm{x}\text{is zero when}\mathrm{x}=\frac{-1}{3}\text{or}\mathrm{x}=\frac{3}{2}.\\ \text{Therefore, the zeroes of}6{\mathrm{x}}^{2}-3-7\mathrm{x}\text{are}\frac{-1}{3}\text{and}\frac{3}{2}\text{.}\\ \text{Now,}\\ \text{Sum of zeroes =}\frac{-1}{3}+\frac{3}{2}=\frac{7}{6}=\frac{-(-7)}{6}=\frac{-(\text{Coefficient of x)}}{{\text{Coefficient of x}}^{2}}\\ \text{Product of zeroes =}\frac{-1}{3}\times \frac{3}{2}=\frac{-1}{2}=\frac{\text{Constant term}}{{\text{Coefficient of x}}^{2}}\\ \text{(iv)}\\ 4{\mathrm{u}}^{2}+8\mathrm{u}=4\mathrm{u}(\mathrm{u}+2)\\ \text{So the value of}4\mathrm{u}(\mathrm{u}+2)\text{is zero when}\mathrm{u}=0\text{or}\mathrm{x}=-2.\\ \text{Therefore, the zeroes of}4\mathrm{u}(\mathrm{u}+2)\text{are}0\text{and}-2.\\ \text{Now,}\\ \text{Sum of zeroes =}0-2=\frac{-8}{4}=\frac{-(\text{Coefficient of u)}}{{\text{Coefficient of u}}^{2}}\\ \text{Product of zeroes =}0\times (-2)=0=\frac{0}{4}=\frac{\text{Constant term}}{{\text{Coefficient of u}}^{2}}\\ \text{(v)}\\ \text{\hspace{0.17em}}{\mathrm{t}}^{2}-15=\text{}{\mathrm{t}}^{2}-{\left(\sqrt{15}\right)}^{2}=\left(\mathrm{t}-\sqrt{15}\right)\left(\mathrm{t}+\sqrt{15}\right)\\ \text{So the value of}{\mathrm{t}}^{2}-15\text{is zero when}\mathrm{t}=\sqrt{15}\text{or}\mathrm{x}=-\sqrt{15}.\\ \text{Therefore, the zeroes of}{\mathrm{t}}^{2}-15\text{are}\sqrt{15}\text{and}-\sqrt{15}\text{.}\\ \text{Now,}\\ \text{Sum of zeroes =}\sqrt{15}-\sqrt{15}=0=\frac{0}{1}=\frac{-(\text{Coefficient of t)}}{{\text{Coefficient of t}}^{2}}\\ \text{Product of zeroes =}\sqrt{15}\times (-\sqrt{15})=-15=\frac{-15}{1}=\frac{\text{Constant term}}{{\text{Coefficient of t}}^{2}}\\ \text{(vi)}\\ \text{}3{\mathrm{x}}^{2}-\mathrm{x}-4=3{\mathrm{x}}^{2}-4\mathrm{x}+3\mathrm{x}-4\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\mathrm{x}(3\mathrm{x}-4)+1(3\mathrm{x}-4)=(3\mathrm{x}-4)(\mathrm{x}+1)\\ \text{So the value of}3{\mathrm{x}}^{2}-\mathrm{x}-4\text{is zero when}\mathrm{x}=\frac{4}{3}\text{or}\mathrm{x}=-1.\\ \text{Therefore, the zeroes of}3{\mathrm{x}}^{2}-\mathrm{x}-4\text{are}\frac{4}{3}\text{and}-1.\\ \text{Now,}\\ \text{Sum of zeroes =}\frac{4}{3}-1=\frac{1}{3}=\frac{-(-1)}{3}=\frac{-(\text{Coefficient of x)}}{{\text{Coefficient of x}}^{2}}\\ \text{Product of zeroes =}\frac{4}{3}\times (-1)=\frac{-4}{3}=\frac{\text{Constant term}}{{\text{Coefficient of x}}^{2}}\end{array}$

**Q.2 **

$\begin{array}{l}\text{Find a quadratic polynomial each with the given}\\ \text{numbers as the sum and product of its zeroes}\\ \text{respectively}.\\ \text{(i)}\frac{1}{4},\text{}-1\text{(ii)}\sqrt{2},\text{\hspace{0.17em}\hspace{0.17em}}\frac{1}{3}\text{(iii) 0,}\sqrt{5}\\ \text{(iv) 1, 1 (v)}\frac{-1}{4},\text{}\frac{1}{4}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(vi) 4, 1}\end{array}$

**Ans.**

$\begin{array}{l}\text{(i)}\frac{1}{4},\text{}-1\\ {\text{Let the quadratic polynomial be ax}}^{2}+\mathrm{bx}+\mathrm{c}\text{and its}\\ \text{zeroes be}\mathrm{\alpha}\text{and}\mathrm{\beta}\text{. Then, we have}\\ \mathrm{\alpha}+\mathrm{\beta}=\frac{1}{4}=\frac{-\mathrm{b}}{\mathrm{a}}\text{and}\mathrm{\alpha \beta}=-1=\frac{\mathrm{c}}{\mathrm{a}}\\ \text{and thus}\\ \mathrm{a}=4,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{b}=-1,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{c}=-4.\\ \text{So, the quadratic polynomial which fits the given}\\ {\text{conditions is 4x}}^{2}-\mathrm{x}-4.\\ \text{(ii)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\sqrt{2},\text{\hspace{0.17em}\hspace{0.17em}}\frac{1}{3}\text{}\\ {\text{Let the quadratic polynomial be ax}}^{2}+\mathrm{bx}+\mathrm{c}\text{and its}\\ \text{zeroes be}\mathrm{\alpha}\text{and}\mathrm{\beta}\text{. Then, we have}\\ \mathrm{\alpha}+\mathrm{\beta}=\sqrt{2}=\frac{-\mathrm{b}}{\mathrm{a}}\text{and}\mathrm{\alpha \beta}=\frac{1}{3}=\frac{\mathrm{c}}{\mathrm{a}}\\ \text{Now,}\\ \mathrm{\alpha}+\mathrm{\beta}=\sqrt{2}=\frac{3\sqrt{2}}{3}=\frac{-\mathrm{b}}{\mathrm{a}}\text{}\\ \text{and}\mathrm{\alpha \beta}=\frac{1}{3}=\frac{\mathrm{c}}{\mathrm{a}}\\ \mathrm{So},\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{a}=3,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{b}=-3\sqrt{2},\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{c}=1.\\ \text{So, the quadratic polynomial which fits the given}\\ {\text{conditions is 3x}}^{2}-3\sqrt{2}\mathrm{x}+1.\\ \text{(iii) 0,}\sqrt{5}\\ {\text{Let the quadratic polynomial be ax}}^{2}+\mathrm{bx}+\mathrm{c}\text{and its}\\ \text{zeroes be}\mathrm{\alpha}\text{and}\mathrm{\beta}\text{. Then, we have}\\ \mathrm{\alpha}+\mathrm{\beta}=0=\frac{-\mathrm{b}}{\mathrm{a}}\text{and}\mathrm{\alpha \beta}=\sqrt{5}=\frac{\mathrm{c}}{\mathrm{a}}\\ \mathrm{So},\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{a}=1,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{b}=0,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{c}=\sqrt{5}.\\ \text{So, the quadratic polynomial which fits the given}\\ {\text{conditions is x}}^{2}+\sqrt{5}.\\ \text{(iv) 1, 1}\\ {\text{Let the quadratic polynomial be ax}}^{2}+\mathrm{bx}+\mathrm{c}\text{and its}\\ \text{zeroes be}\mathrm{\alpha}\text{and}\mathrm{\beta}\text{. Then, we have}\\ \mathrm{\alpha}+\mathrm{\beta}=1=\frac{-\mathrm{b}}{\mathrm{a}}\text{and}\mathrm{\alpha \beta}=1=\frac{\mathrm{c}}{\mathrm{a}}\\ \mathrm{So},\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{a}=1,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{b}=-1,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{c}=1.\\ \text{So, the quadratic polynomial which fits the given}\\ {\text{conditions is x}}^{2}-\mathrm{x}+1.\\ \\ \text{(v)}\frac{-1}{4},\text{}\frac{1}{4}\text{}\\ {\text{Let the quadratic polynomial be ax}}^{2}+\mathrm{bx}+\mathrm{c}\text{and its}\\ \text{zeroes be}\mathrm{\alpha}\text{and}\mathrm{\beta}\text{. Then, we have}\\ \mathrm{\alpha}+\mathrm{\beta}=\frac{-1}{4}=\frac{-\mathrm{b}}{\mathrm{a}}\text{and}\mathrm{\alpha \beta}=\frac{1}{4}=\frac{\mathrm{c}}{\mathrm{a}}\\ \mathrm{So},\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{a}=4,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{b}=1,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{c}=1.\\ \text{So, the quadratic polynomial which fits the given}\\ {\text{conditions is 4x}}^{2}+\mathrm{x}+1.\\ \text{(vi) 4, 1}\\ {\text{Let the quadratic polynomial be ax}}^{2}+\mathrm{bx}+\mathrm{c}\text{and its}\\ \text{zeroes be}\mathrm{\alpha}\text{and}\mathrm{\beta}\text{. Then, we have}\\ \mathrm{\alpha}+\mathrm{\beta}=4=\frac{-\mathrm{b}}{\mathrm{a}}\text{and}\mathrm{\alpha \beta}=1=\frac{\mathrm{c}}{\mathrm{a}}\\ \mathrm{So},\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{a}=1,\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{b}=-4,\text{\%E}\end{array}$

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The NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 can be very helpful for the CBSE board examination preparation of students. The curriculum of NCERT is prepared from the CBSE board examination perspective, which makes it crucial for students to refer to the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 for their preparation. It is recommended for students to refer to the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 as they can benefit students to have a clear understanding of Mathematics topics. To be able to perform well in the CBSE Board Examination of Mathematics, students require consistent practice. Students can utilize the NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 for practice.

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### 3. What are the topics that are covered in Mathematics Class 10 chapter 2 Polynomials?

The topics covered in Mathematics Class 10 chapter 2 Polynomials are mentioned below:-

- What are polynomials
- Degree of a polynomial
- Value of a polynomial
- Zero of polynomial
- Graph of a polynomial
- Zeros of a quadratic polynomial

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