NCERT Solutions For Class 9 Maths Chapter 2 Polynomials (Ex 2.3) Exercise 2.3
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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:
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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.
$\begin{array}{l}\left(\mathrm{iv}\right)\mathrm{}\text{Let P}\left(\text{x}\right)=\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{3}{\mathrm{x}}^{2}(2+\sqrt{2})\mathrm{x}+\sqrt{2}\text{is divided by x}+\text{1 i}.\text{e}.,\text{x =}\u2013\text{1 if x + 1 = 0}.\\ \text{So},\text{P}(\u2013\text{1})\text{}={(\u2013\text{1})}^{3}{(\u2013\text{1})}^{2}(2+\sqrt{2})(\u2013\text{1})+\sqrt{2}\\ \text{\hspace{0.17em}\hspace{0.17em}}=\text{1}\text{1}+\text{2}+\sqrt{2}\text{}+\text{}\sqrt{2}\text{}\\ \text{\hspace{0.17em}\hspace{0.17em}}=\text{2}\sqrt{2}\ne 0\\ \text{Since},\text{remainder is 2}\sqrt{2}.\text{So},\text{}(\text{x}+\text{1})\text{is not a factor of}\mathrm{}\mathrm{the}\text{given polynomial}.\end{array}$
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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.