NCERT Solutions For Class 9 Maths Chapter 2 Polynomials (Ex 2.3) Exercise 2.3
Home > NCERT Solutions > NCERT Solutions For Class 9 Maths Chapter 2 Polynomials (Ex 2.3) Exercise 2.3

CBSE Important Questions›

CBSE Previous Year Question Papers›
 CBSE Previous Year Question Papers
 CBSE Previous Year Question Papers Class 12
 CBSE Previous Year Question Papers Class 10

CBSE Revision Notes›

CBSE Syllabus›

CBSE Extra Questions›

CBSE Sample Papers›
 CBSE Sample Papers
 CBSE Sample Question Papers For Class 5
 CBSE Sample Question Papers For Class 4
 CBSE Sample Question Papers For Class 3
 CBSE Sample Question Papers For Class 2
 CBSE Sample Question Papers For Class 1
 CBSE Sample Question Papers For Class 12
 CBSE Sample Question Papers For Class 11
 CBSE Sample Question Papers For Class 10
 CBSE Sample Question Papers For Class 9
 CBSE Sample Question Papers For Class 8
 CBSE Sample Question Papers For Class 7
 CBSE Sample Question Papers For Class 6

ISC & ICSE Syllabus›

ICSE Question Paper›
 ICSE Question Paper
 ISC Class 12 Question Paper
 ICSE Class 10 Question Paper

ICSE Sample Question Papers›
 ICSE Sample Question Papers
 ISC Sample Question Papers For Class 12
 ISC Sample Question Papers For Class 11
 ICSE Sample Question Papers For Class 10
 ICSE Sample Question Papers For Class 9
 ICSE Sample Question Papers For Class 8
 ICSE Sample Question Papers For Class 7
 ICSE Sample Question Papers For Class 6

ICSE Revision Notes›
 ICSE Revision Notes
 ICSE Class 9 Revision Notes
 ICSE Class 10 Revision Notes

ICSE Important Questions›

Maharashtra board›

RajasthanBoard›
 RajasthanBoard

Andhrapradesh Board›
 Andhrapradesh Board
 AP Board Sample Question Paper
 AP Board syllabus
 AP Board Previous Year Question Paper

Telangana Board›

Tamilnadu Board›

NCERT Solutions Class 12›
 NCERT Solutions Class 12
 NCERT Solutions Class 12 Economics
 NCERT Solutions Class 12 English
 NCERT Solutions Class 12 Hindi
 NCERT Solutions Class 12 Maths
 NCERT Solutions Class 12 Physics
 NCERT Solutions Class 12 Accountancy
 NCERT Solutions Class 12 Biology
 NCERT Solutions Class 12 Chemistry
 NCERT Solutions Class 12 Commerce

NCERT Solutions Class 10›

NCERT Solutions Class 11›
 NCERT Solutions Class 11
 NCERT Solutions Class 11 Statistics
 NCERT Solutions Class 11 Accountancy
 NCERT Solutions Class 11 Biology
 NCERT Solutions Class 11 Chemistry
 NCERT Solutions Class 11 Commerce
 NCERT Solutions Class 11 English
 NCERT Solutions Class 11 Hindi
 NCERT Solutions Class 11 Maths
 NCERT Solutions Class 11 Physics

NCERT Solutions Class 9›

NCERT Solutions Class 8›

NCERT Solutions Class 7›

NCERT Solutions Class 6›

NCERT Solutions Class 5›
 NCERT Solutions Class 5
 NCERT Solutions Class 5 EVS
 NCERT Solutions Class 5 English
 NCERT Solutions Class 5 Maths

NCERT Solutions Class 4›

NCERT Solutions Class 3›

NCERT Solutions Class 2›
 NCERT Solutions Class 2
 NCERT Solutions Class 2 Hindi
 NCERT Solutions Class 2 Maths
 NCERT Solutions Class 2 English

NCERT Solutions Class 1›
 NCERT Solutions Class 1
 NCERT Solutions Class 1 English
 NCERT Solutions Class 1 Hindi
 NCERT Solutions Class 1 Maths

JEE Main Question Papers›

JEE Main Syllabus›
 JEE Main Syllabus
 JEE Main Chemistry Syllabus
 JEE Main Maths Syllabus
 JEE Main Physics Syllabus

JEE Main Questions›
 JEE Main Questions
 JEE Main Maths Questions
 JEE Main Physics Questions
 JEE Main Chemistry Questions

JEE Main Mock Test›
 JEE Main Mock Test

JEE Main Revision Notes›
 JEE Main Revision Notes

JEE Main Sample Papers›
 JEE Main Sample Papers

JEE Advanced Question Papers›

JEE Advanced Syllabus›
 JEE Advanced Syllabus

JEE Advanced Mock Test›
 JEE Advanced Mock Test

JEE Advanced Questions›
 JEE Advanced Questions
 JEE Advanced Chemistry Questions
 JEE Advanced Maths Questions
 JEE Advanced Physics Questions

JEE Advanced Sample Papers›
 JEE Advanced Sample Papers

NEET Eligibility Criteria›
 NEET Eligibility Criteria

NEET Question Papers›

NEET Sample Papers›
 NEET Sample Papers

NEET Syllabus›

NEET Mock Test›
 NEET Mock Test

NCERT Books Class 9›
 NCERT Books Class 9

NCERT Books Class 8›
 NCERT Books Class 8

NCERT Books Class 7›
 NCERT Books Class 7

NCERT Books Class 6›
 NCERT Books Class 6

NCERT Books Class 5›
 NCERT Books Class 5

NCERT Books Class 4›
 NCERT Books Class 4

NCERT Books Class 3›
 NCERT Books Class 3

NCERT Books Class 2›
 NCERT Books Class 2

NCERT Books Class 1›
 NCERT Books Class 1

NCERT Books Class 12›
 NCERT Books Class 12

NCERT Books Class 11›
 NCERT Books Class 11

NCERT Books Class 10›
 NCERT Books Class 10

Chemistry Full Forms›
 Chemistry Full Forms

Biology Full Forms›
 Biology Full Forms

Physics Full Forms›
 Physics Full Forms

Educational Full Form›
 Educational Full Form

Examination Full Forms›
 Examination Full Forms

Algebra Formulas›
 Algebra Formulas

Chemistry Formulas›
 Chemistry Formulas

Geometry Formulas›
 Geometry Formulas

Math Formulas›
 Math Formulas

Physics Formulas›
 Physics Formulas

Trigonometry Formulas›
 Trigonometry Formulas

CUET Admit Card›
 CUET Admit Card

CUET Application Form›
 CUET Application Form

CUET Counselling›
 CUET Counselling

CUET Cutoff›
 CUET Cutoff

CUET Previous Year Question Papers›
 CUET Previous Year Question Papers

CUET Results›
 CUET Results

CUET Sample Papers›
 CUET Sample Papers

CUET Syllabus›
 CUET Syllabus

CUET Eligibility Criteria›
 CUET Eligibility Criteria

CUET Exam Centers›
 CUET Exam Centers

CUET Exam Dates›
 CUET Exam Dates

CUET Exam Pattern›
 CUET Exam Pattern
The Central Board of Secondary Education (CBSE), a national secondary education governing body with its main office in New Delhi, provides services to both public and private schools. The Government of India is in charge of the CBSE Board, which is under its control. To advance to Class 12, students in Class 11 must receive at least a 33% on their theory and practical exams. Students who don’t score well in a particular subject may take a compartment exam. Students who fail more than two topics or a compartment must repeat them all the following year. The National Council of Educational Research and Training, a government agency, is in charge of raising the standard of schooling in India. NCERT must be supervised by the CBSE Board. The National Council for Educational Research and Training (NCERT) was created by the Indian Government in 1961 to offer guidance and assistance to the Central Government and State Governments. The National Council for Educational Research and Training (NCERT) creates and publishes model textbooks, supplemental materials, newsletters, journals, educational kits, and multimedia digital goods.
The NCERT books are sufficient to cover the whole curriculum offered by the CBSE, but they also cover all the foundational concepts of every subject taught in CBSEaffiliated schools as well as other schools, clearly and straightforwardly. This aids students in clarifying their understanding of various ideas. Students do not need to repeatedly cram the study material once a subject has been understood. Students only need to revise theories, formulas, equations, etc. and how they are applied throughout the exam. Due to this, the NCERT books are regarded as the ideal resource for full and extensive preparation that can aid in the development of distinct conceptions.
NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 (Ex 2.3) (Include PDF)
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, are available at Extramarks. This chapter deals with polynomials. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, shows how to solve problems using Polynomial long division. Extramarks offers downloadable solutions for all NCERT Solutions for Class 9 Maths Chapter 2 Exercise 2.3 that have been updated in CBSE textbooks. Subjects such as Science, Mathematics, and English are easier to learn when students have access to NCERT Class 9 Solutions, NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, and other subject solutions available exclusively on Extramarks.
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 for Class 9 Maths Chapter 2 Exercise 2.3 covers the basics of Polynomials, including Various types of Polynomials, finding roots and solving Polynomials. A Polynomial is an algebraic expression with one or more variables. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, also cover the Remainder Theorem, Polynomial Factor Theory, Algebraic Identities, and Polynomials of Various Degrees. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, shows the difference between 1st, 2nd, and 3rddegree Polynomials. The Remainder Theorem and the Factor Theorem are the two important sets that help identify the factors of Polynomials.
Access NCERT Solutions For Class 9 Mathematics Chapter 2 – Polynomials
Passing the CBSE exam requires good preparation and dedication. This can be achieved with the right amount of hard work and consistent practice. Working smartly and strategically in preparation is also very important. After studying this chapter, solving Mathematics problems with NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 will help students understand the pattern of the exam and the weightage of each topic. Students should remember that it is important to speed up their preparation during the exam. This is the point at which many students fail to finish their assignments on time. By practising NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 they will acquire a tendency to think about answers to given problems at breakneck speed. There are many terms in Chapter 2 of Mathematics which are discussed in NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 for Class 9 Maths Ex 2.3.
Variables And Constants
The study material available on the Extramarks website is wellorganized and easy to understand. NCERT Solutions is just one of the many examples of this. The different types of tools available on the Extramarks website are all organised in a chapterwise format, which makes it easier for the students to not only find them easily but also saves a lot of their time and energy. Students can go through the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 for a thorough understanding of variable and constants.
Terms
All the resources available for the students are prepared in a way that they are easily accessible and the students can get them with a click. This saves the students a lot of time that they can now put to use in preparing for their examinations. This helps the students not get overwhelmed or stressed by the situation and ultimately perform better in their examinations. All the terms have been explained in the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3.
Polynomial
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, can give the students an allaround understanding of the topics covered in this particular chapter. These solutions are easy to understand, so the students can make use of them for their own studies and make the most of them.
Examples Of Polynomials
The NCERT solutions that are available on the Extramarks website can give the students an indepth understanding of all the important topics and concepts. The solutions are provided with various examples that can help the students understand the topics better.
What You Will Learn In Class 9 Maths Chapter 2 Exercise 2.3
In NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 students will learn how to divide a Polynomial by another Polynomial or any term. Class 9 Mathematics Chapter 2 Exercise 2.3 is useful for finding division remainders and also indicates whether a Polynomial is part of another Polynomial.
NCERT Solutions Class 9 Maths Chapter 2 Exercises
There is always time pressure when preparing for exams. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 is provided as a study resource to assist students in quickly and systematically reviewing the chapter during exam preparation. It does not take long and covers the entire curriculum. Practising NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 will help students correct their mistakes and ease the pain of solving the chapter because speeding up preparation is very important.
Exercise 2.3 Class 9 Math act as revision notes because students have the opportunity to review all the chapters they have studied so far in a Q&A format. This is actually more beneficial than reading notes. Repeating questionandanswer patterns can help them capture or remember key points they may have missed while reading a chapter. That is, brushing and summarising the chapters for easier memorisation. So they should take a look at NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, and try to improve their weak areas by correcting the mistakes.
Analyze and strategize the preparation. Testing the knowledge helps them understand and reflect on how wellprepared they are. Often, students study the entire chapter well and still cannot answer the questions. This can be true for a variety of reasons, such as lack of practice, repetition, or missing important points while reading the chapter superficially. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, will help them strategize the preparation to get the maximum score from what they have already learned.
Interpretation and level of complexity of key questions: Although CBSE is changing the pattern of questionnaires in exams, it is important to realize that the chapter concepts and topics remain the same. Answering the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 will help them identify the very important themes for that chapter. Additionally, they can find out the level of complexity in the questions that will be asked during the board exam and prepare accordingly.
Improve speed through practice. Practice makes students confident in what they learn and gives them the ability to write perfect answers from the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, which can be helpful. Extramarks always recommends that students keep practising important topics and questions so that they do not forget them during the exam. Solving NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, involves answering the same question in different ways, ultimately leading to concept retention and speed.
The more problems they solve, the better they will understand the techniques used in preparing questions. They can check the number of repetitive questions and their interval. Gauging NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 will make them more likely to perform well on their next exam as they will notice topics that have not appeared on the exam for a long time. This way they can infer important themes without going through the exact same questions, but the concept remains the same.
Extramarks’ NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, helps them prepare well. Extramarks strongly believes in intellectual work and embraces experienced teachers who are adept at learning and passionate about teaching the same. Extramarks make the learning experience fun by providing Class 9 Mathematics Chapter 2 Exercise 2.3 Solutions with stepbystep explanations of numerical problems that help deepen the understanding of concepts related to the topic. Developed by Extramarks experts, the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 serves as great material for practice and makes the learning process more convenient.
By enabling NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, Extramarks encourages the smartest way to learn with the following benefits:
 Wellstandardized solutions contain comprehensive answers to all questions.
 Comprehensive, stepbystep solutions.
 Carefully compiled by a professional.
 Helps keep the topics under retention.
 Boosts confidence.
 Written and reviewed by experienced subjectmatter experts.
 It has been revised in line with the latest NCERT syllabus and the guidelines of the Central Board of Secondary Education (CBSE).
 Saves time.
 Serves as good revision notes.
 Helps pass various competitive entrance exams.
 It is written with the age group of the students in mind.
 The solutions are written in plain language, emphasizing basic facts, terms, principles, and application of various concepts.
 Complicated solutions are broken down into simpler parts, reducing unnecessary mental strain on students.
 Provides an overview and concepts for the entire chapter in the form of a solution.
 Responses are systematically processed and presented in a coherent and interesting way.
 The content is concise, short and selfexplanatory.
 Some answers contain the necessary images to help understand the concepts.
 Extramarks NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 solutions keep up with the latest curriculum and exam regulations.
NCERT Solutions For Class 9
Students can find the solutions to every kind of problem they face with a proper explanation.
There are six sections and five exercises in the Polynomial chapter of Class 9. However, the basic concepts in NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, revolve around unknown variables, calculation, and finding a solution to unknown variables. But the techniques used can be different. Practice is the key to getting the best scores on exams. However, for that, one needs proper solutions to check the steps. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3, can help students get the best results and outcomes.
CBSE Study Materials For Class 9
Extramarks has done its best to provide real assistance with the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 for Class 9 Polynomials. The aim was to provide enough problems and solutions for practice and to provide a strong foundation for the chapters.
There are a total of four exercises in Class 9 Polynomials, the last one being optional. For the first exercise, students need to find the roots of the polynomial p(x). For the second exercise, they have two questions. The first requires checking the relationship between the zeros and the coefficients, and the second requires finding the quadratic polynomial. The third problem has a total of 5 problems where they need to split the polynomial and find the root of the polynomial. Then there is an optional 5question exercise where they need to find the roots of a polynomial.
A compact exercise follows each topic. This exercise aims to test your knowledge and deep understanding of the various theorems and concepts presented in this chapter. Nevertheless, it should be noted that the numerical problems in this chapter are mainly based on certain theorems and other related concepts. Numerous examples of solving numerical problems are also provided to deepen the understanding of these topics and related concepts. Additionally, detailed stepbystep instructions are provided for each solved example. It helps to understand the methods used to tackle different types of questions in order to solve them accurately.
CBSE Study Materials
There are many benefits of using the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 before the exam. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 from Extramarks are designed to provide basic understanding and simple and easy solutions. Students feel relaxed after being presented with individual solution steps and correct explanations. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 are always in demand for CBSE students. The PDF of NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 are available on Extramarks and contains many preanswered questions of different types, so students will not feel exhausted on the last attempt of the exam. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 covers the entire syllabus and includes good stepbystep instructions. Emphasis is also placed on developing students’ troubleshooting and problemsolving skills. Taking the exam is a distant concept. The basis is learning and understanding. It is important to avoid rushing Mathematics exam preparation. So, NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 will definitely help them improve their grades.
Q.1 Find the remainder when x^{3} – ax^{2} + 6x – a is divided by x – a.
Ans
{\text{x}}^{\text{3}}\u2013{\text{ax}}^{\text{2}}+\text{6x}\u2013\text{a is divided by x}\u2013\text{a}
\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}^{2}+6\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{x}\u2013\text{a}\overline{){\text{x}}^{\text{3}}\u2013{\text{ax}}^{\text{2}}+\text{6x}\u2013\text{a}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{\xaf}{\pm {x}^{3}\mp a{x}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6xa\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{\xaf}{\text{\hspace{0.17em}}\pm 6x\mp 6a}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{\xaf}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}5a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\\ \end{array}
T
$\begin{array}{l}Another\text{method:}\\ \text{Let}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{p}\left(x\right)={\text{x}}^{\text{3}}\u2013{\text{ax}}^{\text{2}}+\text{6x}\u2013\text{a and the zero of x}\u2013\text{a is a}\text{.}\\ \text{So, p}\left(a\right)={\left(a\right)}^{\text{3}}\u2013{\left(a\right)}^{\text{3}}+\text{6}\left(a\right)\u2013\text{a}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=5a\\ \text{Thus, the remainder is}5a\text{.}\end{array}$
Q.2 Check whether 7 + 3x is a factor of 3x^{3} + 7x.
Ans
$\begin{array}{l}\text{Let}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{p}\left(x\right)={\text{3x}}^{\text{3}}+\text{7x is divided by 7}+\text{3x i}\text{.e}\text{., x=}\frac{7}{3}\mathrm{if}7+3\mathrm{x}=0\text{.}\\ \text{So, p}(\frac{7}{3})=\text{3}{(\frac{7}{3})}^{\text{3}}+\text{7}(\frac{7}{3})\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{343}{9}\frac{49}{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{343147}{9}=\frac{490}{9}\\ \text{Thus, the remainder is}\frac{490}{9}\text{.}\\ \mathrm{Therefore},\text{7}+{\text{3x is not a factor of 3x}}^{\text{3}}+\text{7x}.\end{array}$
Q.3
$\begin{array}{l}\text{Determine which of the following polynomials has}\left(\text{x+1}\right)\text{a}\\ \text{factor:}\\ \left(\text{i}\right){\text{\hspace{0.33em}x}}^{\text{3}}{\text{+x}}^{\text{2}}\text{+x+1}\\ \left(\text{ii}\right){\text{\hspace{0.33em}x}}^{\text{4}}{\text{+x}}^{\text{3}}{\text{+x}}^{\text{2}}\text{+x+1}\\ \left(\text{iii}\right){\text{\hspace{0.33em}x}}^{\text{4}}{\text{+3x}}^{\text{3}}{\text{+3x}}^{\text{2}}\text{+x+1}\\ \left(\text{iv}\right){\text{\hspace{0.33em}x}}^{\text{3}}{\text{x}}^{\text{2}}\left(\text{2+}\sqrt{\text{2}}\right)\text{x+}\sqrt{\text{2}}\end{array}$
Ans
(i) Let P(x) = x^{3} + x^{2} + x + 1 is divided by x+1 i.e., x = –1 if x + 1 = 0.
So, P(–1) = (–1)^{3} + (–1)^{2} + (–1) + 1
= –1 + 1 –1 + 1 = 0
Since, remainder is 0. So, (x + 1) is a factor of the given polynomial.
(ii) Let P(x) = x^{4} + x^{3} + x^{2} + x + 1 is divided by x+1
i.e., x = –1 if x + 1 = 0.
So, P(–1) = (–1)^{4} + (–1)^{3} + (–1)^{2} + (–1) + 1
= 1 – 1 + 1 – 1 + 1 = 1
Since, remainder is 1. So, (x + 1) is not a factor of the given polynomial.
(iii) Let P(x) = x^{4} + 3x^{3} + 3x^{2} + x + 1 is divided by x+1
i.e., x = –1 if x + 1 = 0.
So, P(–1) = (–1)^{4} + 3(–1)^{3} + 3(–1)^{2} + (–1) + 1
= 1 – 3 + 3 – 1 + 1 = 1
Since, remainder is 1. So, (x + 1) is not a factor of the given polynomial.
$\begin{array}{l}\left(\mathrm{iv}\right)\mathrm{}\text{Let P}\left(\text{x}\right)=\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{3}{\mathrm{x}}^{2}(2+\sqrt{2})\mathrm{x}+\sqrt{2}\text{is divided by x}+\text{1 i}.\text{e}.,\text{x =}\u2013\text{1 if x + 1 = 0}.\\ \text{So},\text{P}(\u2013\text{1})\text{}={(\u2013\text{1})}^{3}{(\u2013\text{1})}^{2}(2+\sqrt{2})(\u2013\text{1})+\sqrt{2}\\ \text{\hspace{0.17em}\hspace{0.17em}}=\text{1}\text{1}+\text{2}+\sqrt{2}\text{}+\text{}\sqrt{2}\text{}\\ \text{\hspace{0.17em}\hspace{0.17em}}=\text{2}\sqrt{2}\ne 0\\ \text{Since},\text{remainder is 2}\sqrt{2}.\text{So},\text{}(\text{x}+\text{1})\text{is not a factor of}\mathrm{}\mathrm{the}\text{given polynomial}.\end{array}$
Please register to view this section
FAQs (Frequently Asked Questions)
1. What is a Polynomial?
A Polynomial is an expression that consists of variables (or indefinite), terms, exponents, and constants. For example, 3×2 2x10 is a Polynomial. Students can go through the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.3 for a thorough understanding of Polynomials.
2. What is the degree of the Zero Polynomial and the Constant Polynomial?
The degree of the Zero Polynomial is undefined, but the degree of the Constant Polynomial is one.
3. How can students boost their Class 10 board exam preparation?
It is crucial to have a comprehensive comprehension of the complete course content in Mathematics in order to adequately prepare for the NCERT Class 11 Examination. Students should finish the full course at least a month before the start of Class 11 final exams. The aim of Extramarks is to guarantee that students can effectively learn and review material. Prior to the exam, it is advised that candidates become familiar with the questions from past years’ papers and practice tests. Students’ confidence is raised when they successfully respond to questions, which enhances their performance on tests.