NCERT Solutions For Class 9 Maths Chapter 2 Polynomials (Ex 2.1) Exercise 2.1
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Polynomials have great importance in the languages of Mathematics and Algebra. They are used in almost all areas of Mathematics to represent numbers as the results of Mathematical operations. Polynomials are also “building blocks” for other types of mathematical expressions, such as Reasonable Expressions.
Many mathematical processes performed in everyday life can be interpreted as polynomials. The total cost of items on a grocery bill can be interpreted as a polynomial. Calculating the distance travelled by a vehicle or object can be interpreted as a polynomial. Calculations of the perimeter, area, and volume of geometric figures can be interpreted as polynomials. These are just a few of the many uses of polynomials.
NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 (Ex 2.1) (Include PDF)
The second chapter of the Class 9 Mathematics NCERT book is Polynomials. Building on the foundations learned in Chapter 1, Number System, Polynomials is another important chapter in the Class 9 curriculum that is relevant for higher classes. A polynomial is an algebraic expression containing different terms and having variables of different powers.
Here are some ways to study Polynomials:
Students can first look at NCERT –
To understand the basics of the chapters on the first read, it is best to read the chapter on polynomials from NCERT rather than any other book. The way polynomials are structured in NCERT books is very good for understanding the chapter from the beginning. It will be important to first explain the definition and then give examples to extend the definition later with more complex topics based on what students have learned to understand the concept.
Students should understand key concepts and definitions –
To understand the Polynomials chapter, it is important to understand the key concepts and definitions presented in this chapter. Variables, constants, coefficients, zero polynomials, factors, polynomial roots, etc. These concepts are important for understanding the topics later in this chapter and for understanding the basics.
Students should not ignore the theory given by NCERT –
The Polynomial chapter also has a theory that introduces students to the chapter and the important issues in that chapter. The theory explains these definitions and topics in a very effortless way, allowing reading students to quickly understand the topic. The theory attempts to explain concepts in innovative and interesting ways. This keeps students engaged while reading and explains concepts very easily
Students should note the resolved example –
After students have learned the concepts and foundations of the chapter, they should focus on the examples presented after the concepts are explained and the problems based on the concepts just explained. These examples help students interpret the theory they read about the subject by providing different questions that can be asked based on that subject, and explaining how to solve those questions to help students understand the logic and problems. Doing so improves their problemsolving abilities. Some of the solved examples come directly from previous exams, showing how important it is to pay attention to these examples.
Students must address the question asked in the exercise –
After working through examples and learning how to solve problems, students are required to solve problems given in various exercises in the Polynomial chapter. Solving these questions and checking the correctness of their answers on the back cover of the book will give students immense confidence in the subject at hand and increase their interest in the chapter.
For more clarity, students should practice with the help of NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1. Once the student has a thorough understanding of the NCERT in Polynomials, they should refer to NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 to further practice the problems in that chapter. Solving more problems reveals a wider variety of problems based on the chapters about them, so students will know how to try out different types of problems beforehand. If they cannot solve the problem given for polynomials, they can refer to NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1.
Students must develop an understanding of Key Phrases: Remainders and Factors
In the Polynomials chapter, some important theorems are given that are essential from the point of view of the exam: the Remainder Theorem and the Factor Theorem. These theorems may be difficult to understand at first, but students should listen to the explanations and try to understand them with examples. These topics often result in highlyrated questions on exams. So a thorough understanding of these theorems is essential.
Students must remember the various algebraic identities given:
The algebraic identities described in this chapter are often used with polynomials. These algebraic identities make it easy to solve some problems that otherwise might have required more work. An algebraic identity is basically an algebraic expression that is true for all values of the variables present in the expression. Therefore, all students should be able to memorise these IDs and remember them during exams.
Students should note down the important formulas:
For the important formulas and identities given in the polynomial chapter, students should write them down in a separate notebook so that they can review these identities and formulas whenever they have time.
Students should see the summary at the end of the chapter:
After the chapter is over, the NCERT book has a chapter summary, summarising all the important definitions and formulas students learned in polynomials. So, after reading this chapter, students should look at this summary to help them remember what they learned in this chapter.
Access NCERT Solutions For Class 9 Mathematics Chapter 2 – Polynomials
Polynomials are an important topic in the Class 9 Mathematics Syllabus, and therefore, it is essential for students to study this chapter with concentration. Polynomials are covered in the Class 9 Maths Chapter 2 Exercise 2.1 Mathematics syllabus. Exercise 2.1 Class 9 Math is a very important exercise in the chapter covered in Class 9, which is divided into eight major themes. Students are encouraged to practice the problems they solve so that they can master the concept of Polynomials.
A polynomial is an expression that has three or more algebraic terms. They are also defined as the sum of multiple terms with the same variable with different powers. Polynomials are actually the language used in most mathematical formulas to represent the appropriate relationships between various variables or numbers. Students are encouraged to collect as much information as possible from Exercise 2.1 Class 9 Maths Solution to help them easily solve the tricky polynomialbased problems on the exam.
Important topics in Polynomials Class 9 Exercise 2.1:
Definition of a Polynomial in one variable; its coefficients; examples and counterexamples; its terms; the Zero Polynomial; Degree of the Polynomial; Constant, Linear, Quadratic, And Cubic Polynomials; Monomials, Binomials, and Trinomials; Factors and Multiples; and Zeros/Roots of Polynomials/Equations. Students must formulate and justify the Remainder Theorem with examples and Integer analogies. Description and proof of a set of factors. Factorization of ax2 + bx + c, a ≠ 0, where a, b, and c are real numbers, and factorization of cubic Polynomials using coefficient sets.
What Do You Learn From The Class 9 Maths Chapter 2 Exercise 2.1?
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 introduces Polynomials. Section 2.1 discusses Polynomials, how algebraic expressions are constructed, and how to use exponential powers represented by equations to test whether a variable is Polynomial. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 is an important part of the CBSE Class 9 Mathematics syllabus, referring to Polynomials makes a significant proportion of the Mathematics syllabus.
NCERT Class 9 Maths Chapter 2 Exercise 2.1
The CBSE Class 9 Mathematics curriculum finally introduces students to the basics of high school mathematics. Students must know that some of the formulas they learned will be wellsuited for their future careers. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1, presents a series of tricky problems commonly found in CBSE Class 9 Mathematics papers. A student who wants to get the highest marks in the CBSE Mathematics exam in Class 9 must master the subject before the exam date. When it comes to regular practice, it is important to know all the exact steps for deriving Mathematical solutions. In this regard, Extramarks’s PDF of NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 answers will help students greatly improve their polynomial problemsolving skills. The PDF of the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1, is portable in nature and can be carried on smartphones, laptops, etc. and used conveniently to review problemsolving and improve the skills of the students preparing for Class 9 Mathematics.
NCERT Solutions For Class 9 Maths
In addition to the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1, students are also provided with other online learning materials, such as notebooks, books, surveys, example assignments, and worksheets available on Extramarks. These materials were developed in relation to the CBSE syllabus and NCERT curriculum. Students are also encouraged to practice the CBSE Class 9 Sample Papers to get a feel for the pattern of questions on the final exam.
NCERT textbook for Class 9 Mathematics contains 15 chapters. These NCERT Class 9 Mathematics chapters form the basis for the Class 10 chapters. The PDF version of NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 provided by Extramarks is publicly accessible and available for students to download.
Access Other Exercise Solutions Of Class 9 Maths Chapter 2 Polynomials
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1, describes a particular kind of algebraic expression called polynomials and related terminology. A polynomial is an expression composed of variables and coefficients, including addition, subtraction, multiplication, and nonnegative integer exponential operations.
NCERT Solutions For Class 9
Using these theorems in Polynomial factorization, the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 also deal with the Remainder Theorem and the Factor Theorem. Students are taught several examples as well as definitions of various terms such as Polynomials, Degrees, Coefficients, Roots, and Polynomial terms. This chapter contains a total of 5 exercises with problems on all the topics explained in NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1.
CBSE Study Materials For Class 9
Extramarks offers NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1. This is one of the most important chapters in the CBSE Class 9 mathematics syllabus. Most students looking to master their Class 9 Mathematics syllabus must refer to NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1. Students can download Extramarks’ NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 created by experts based on CBSE guidelines. Extramarks provides students with a PDF download option for the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1. Subjects such as science, mathematics, and english will be easier to learn with access to NCERT Class 9 Science, Mathematics solutions and other subject solutions and study materials available on Extramarks.
CBSE Study Materials
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 lay the foundation for higher education. Practising exercises along with NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 clears students’ doubts and makes it easier for students to solve problems through NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1. Students can download a PDF of NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1, Science, Social Studies, English and Hindi.
Q.1 Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
$\begin{array}{l}\left(\text{i}\right){\text{\hspace{0.33em}4x}}^{\text{2}}\text{3x+7}\\ \left(\text{ii}\right){\text{\hspace{0.33em}y}}^{\text{2}}\text{+}\sqrt{\text{2}}\\ \left(\text{iii}\right)\text{\hspace{0.33em}3}\sqrt{\text{t}}\text{+t}\sqrt{\text{2}}\\ \left(\text{iv}\right)\text{\hspace{0.33em}y+}\frac{\text{2}}{\text{y}}\left(\text{v}\right){\text{x}}^{\text{10}}{\text{+y}}^{\text{3}}{\text{+t}}^{\text{50}}\end{array}$
Ans
$\begin{array}{l}\left(\mathrm{i}\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}4{\mathrm{x}}^{2}3\mathrm{x}+7\to \mathrm{It}\text{is a polynomial in variable x.}\\ \left(\mathrm{ii}\right)\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{y}}^{2}+\sqrt{2}\to \text{It is a polynomial in variable y.}\\ \left(\mathrm{iii}\right)\text{\hspace{0.17em}}3\sqrt{\mathrm{t}}+\mathrm{t}\sqrt{2}=3{\mathrm{t}}^{\frac{1}{2}}+\mathrm{t}\sqrt{2}\\ \mathrm{It}\text{is not a polynomial in variable t because power of t is in}\\ \text{fraction i.e.,}\frac{1}{2}.\\ \left(\mathrm{iv}\right)\text{\hspace{0.17em}}\mathrm{y}+\frac{2}{\mathrm{y}}=\mathrm{y}+2{\mathrm{y}}^{1}\\ \mathrm{It}\text{is not a polynomial in variable y because power of y is negative}\\ \text{i.e.,}\text{1.}\\ \left(\mathrm{v}\right){\mathrm{x}}^{10}+{\mathrm{y}}^{3}+{\mathrm{t}}^{50}\to \mathrm{It}\text{is not a polynomial in one variable because}\\ \text{there are three variables i.e., x, y and t.}\end{array}$
Q.2
$\begin{array}{l}{\text{Write\hspace{0.33em}the\hspace{0.33em}coefficients\hspace{0.33em}of\hspace{0.33em}x}}^{\text{2}}\text{\hspace{0.33em}in\hspace{0.33em}each\hspace{0.33em}of\hspace{0.33em}the\hspace{0.33em}following:}\\ \left(\text{i}\right){\text{\hspace{0.33em}2+x}}^{\text{2}}\text{+x}\\ \left(\text{ii}\right)\text{\hspace{0.33em}2}{\text{x}}^{\text{2}}{\text{+x}}^{\text{3}}\\ \left(\text{iii}\right)\text{\hspace{0.33em}}\frac{\text{\pi}}{\text{2}}{\text{x}}^{\text{2}}\text{+x}\\ \left(\text{iv}\right)\text{\hspace{0.33em}}\sqrt{\text{2}}\text{x}\text{1}\end{array}$
Ans
$\begin{array}{l}\left(\mathrm{i}\right)\text{\hspace{0.33em}}2+{\mathrm{x}}^{2}+\mathrm{x}\\ \mathrm{Coefficient}{\text{of x}}^{\text{2}}\text{is 1.}\\ \left(\mathrm{ii}\right)\text{\hspace{0.33em}}2{\mathrm{x}}^{2}+{\mathrm{x}}^{3}\\ \mathrm{Coefficient}{\text{of x}}^{\text{2}}\text{is\hspace{0.17em}}\text{1.}\\ \left(\mathrm{iii}\right)\text{\hspace{0.33em}}\frac{\mathrm{\pi}}{2}{\mathrm{x}}^{2}+\mathrm{x}\\ \mathrm{Coefficient}{\text{of x}}^{\text{2}}\text{is}\frac{\mathrm{\pi}}{2}\text{.}\\ \left(\mathrm{iv}\right)\text{\hspace{0.33em}}\sqrt{2}\mathrm{x}1\\ \mathrm{Coefficient}{\text{of x}}^{\text{2}}\text{is 0.}\end{array}$
Q.3 Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Ans Binomial of degree 35 is x^{35} + 15. Monomial of degree 100 is x^{100}.
Q.4
$\begin{array}{l}\text{Write\hspace{0.33em}the\hspace{0.33em}degree\hspace{0.33em}of\hspace{0.33em}each\hspace{0.33em}of\hspace{0.33em}the\hspace{0.33em}following\hspace{0.33em}polynomials:}\\ \left(\text{i}\right){\text{\hspace{0.33em}5x}}^{\text{3}}{\text{+4x}}^{\text{2}}\text{+7x}\\ \left(\text{ii}\right)\text{\hspace{0.33em}4}{\text{y}}^{\text{2}}\\ \left(\text{iii}\right)\text{\hspace{0.33em}5t}\sqrt{\text{7}}\\ \left(\text{iv}\right)\text{\hspace{0.33em}3}\end{array}$
Ans
$\begin{array}{l}\left(\mathrm{i}\right)\mathrm{}5{\mathrm{x}}^{3}+4{\mathrm{x}}^{2}+7\mathrm{x},\\ \mathrm{The}\mathrm{degree}\mathrm{of}\mathrm{the}\text{given polynomial is 3 becuase the heighest power of x is 3.}\\ \left(\mathrm{ii}\right)\mathrm{}4{\mathrm{y}}^{2},\text{}\\ \text{}\mathrm{The}\mathrm{degree}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{polynomial}\mathrm{is}2\mathrm{becuase}\text{heighest power of y is 2.}\\ \left(\mathrm{iii}\right)\mathrm{}5\mathrm{t}\sqrt{7},\text{\hspace{0.17em}}\\ \mathrm{The}\mathrm{degree}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{polynomial}\mathrm{is}1\mathrm{becuase}\text{heighest power of t is 1.}\\ \left(\mathrm{iv}\right)\mathrm{}3\\ \mathrm{The}\mathrm{degree}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{polynomial}\mathrm{is}0\mathrm{becuase}\text{of constant term.}\end{array}$
Q.5
$\begin{array}{l}\mathrm{Classify}\text{\hspace{0.33em}}\mathrm{the}\text{\hspace{0.33em}}\mathrm{following}\text{\hspace{0.33em}}\mathrm{as}\text{\hspace{0.33em}}\mathrm{linear},\text{\hspace{0.33em}}\mathrm{quadratic}\text{\hspace{0.33em}}\mathrm{and}\text{\hspace{0.33em}}\mathrm{cubic}\\ \mathrm{polynomials}:\\ \left(\mathrm{i}\right)\text{\hspace{0.33em}}{\mathrm{x}}^{2}+\mathrm{x}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\mathrm{ii}\right)\mathrm{x}{\mathrm{x}}^{3}\left(\mathrm{iii}\right)\text{\hspace{0.33em}}\mathrm{y}+{\mathrm{y}}^{2}+4\text{\hspace{0.33em}}\left(\mathrm{iv}\right)\text{\hspace{0.33em}}1+\mathrm{x}\\ \left(\mathrm{v}\right)\text{\hspace{0.33em}}3\mathrm{t}\text{\hspace{0.33em}}\left(\mathrm{vi}\right)\text{\hspace{0.33em}}{\mathrm{r}}^{2}\text{\hspace{0.33em}}\left(\mathrm{vii}\right)\text{\hspace{0.33em}}7{\mathrm{x}}^{3}\end{array}$
Ans
$\begin{array}{l}\left(\mathrm{i}\right)\text{\hspace{0.33em}}{\mathrm{x}}^{2}+\mathrm{x}\\ \mathrm{This}\text{polynomial is quadratic polynomial because\hspace{0.17em}\hspace{0.17em}degree of the}\\ \text{polynomial is 2.}\\ \left(\mathrm{ii}\right)\text{\hspace{0.33em}}\mathrm{x}{\mathrm{x}}^{3}\\ \mathrm{This}\text{polynomial is cubic polynomial because\hspace{0.17em}\hspace{0.17em}degree of the}\\ \text{polynomial is 3.}\\ \left(\mathrm{iii}\right)\text{\hspace{0.33em}}\mathrm{y}+{\mathrm{y}}^{2}+4\\ \mathrm{This}\text{polynomial is quadratic polynomial because\hspace{0.17em}\hspace{0.17em}degree of the}\\ \text{polynomial is 2.}\\ \left(\mathrm{iv}\right)\text{\hspace{0.33em}}1+\mathrm{x}\\ \mathrm{This}\text{polynomial is linear polynomial because\hspace{0.17em}\hspace{0.17em}degree of the}\\ \text{polynomial is 1.}\\ \left(\mathrm{v}\right)\text{\hspace{0.33em}}3\mathrm{t}\\ \mathrm{This}\text{polynomial is linear polynomial because\hspace{0.17em}\hspace{0.17em}degree of the}\\ \text{polynomial is 1.}\\ \left(\mathrm{vi}\right)\text{\hspace{0.33em}}{\mathrm{r}}^{2}\\ \mathrm{This}\text{polynomial is quadratic polynomial because\hspace{0.17em}\hspace{0.17em}degree of the}\\ \text{polynomial is 2.}\\ \left(\mathrm{vii}\right)\text{\hspace{0.33em}}7{\mathrm{x}}^{3}\\ \mathrm{This}\text{polynomial is cubic polynomial because\hspace{0.17em}\hspace{0.17em}its degree is 3.}\end{array}$
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FAQs (Frequently Asked Questions)
1. How can students effectively use the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1?
Algebra forms the basis of higher Mathematics studies. Therefore, students should focus on the key terms defined in this chapter, such as the degree of Polynomials, the difference between constants and variables, etc., for a clear understanding of Polynomials. This will give them a solid foundation for the board exams and face all kinds of tough questions.
2. Why are NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 Important?
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 contain a detailed explanation of the important part of Algebra, remainders and factor sets. Also, important Algebraic Identities are explained in detail and there are many questions for students to solve. A list of all important equations and concepts can be found at the end of the chapter. This is of great advantage as students can use this list while going through the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1. Overall, the NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 cover all important concepts, approaches, and formulas, making NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 the most important thing for Class 9 Mathematics students.
3. Should students practice all the problems in NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1?
The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 contain a variety of problems exploring all algebraic concepts related to Polynomials. Therefore, Extramarks encourages students to use this resource and start practising by solving the examples. This will help them understand the steps they need to follow when solving Polynomial problems. The NCERT Solutions For Class 9 Maths Chapter 2 Exercise 2.1 will help students in solving the questions in a stepbystep manner.