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Class 10 Maths Chapter 2 Solutions Polynomials will discuss about the polynomials with any degree and its solutions using a graphical representation. These solutions are prepared by expert teachers keeping in mind the latest CBSE syllabus 2023. Also, Class 10 maths chapter 2 solutions polynomials gives a detailed explanation of questions given in the NCERT Book of Class 10 Maths.
You must refer to these Polynomial Class 10 notes to check the correct answers and proper explanation of each question. These NCERT Solutions for Class 10 Maths Chapter 2 Polynomials are prepared by experts and are easy understand. You can check the NCERT solutions for Class 10 for other subjects here.
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The general Polynomial Expression is represented as follows:
F(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + rx + s.
For a natural number n, a^{n}-b^{n} = (a - b)(a^{n-1} + a^{n-2}b + ... + b^{n-2}a + b^{n-1})
When n is an even number (expressed as n = 2a),
x^{n} + y^{n} = (x + y)(x^{n-1} - x^{n-2}y + ... + y^{n-2}x - y^{n-1})
For an odd number n,
x^{n} + y^{n} = (x + y)(x^{n-1} - x^{n-2}y + ... - y^{n-2}x + y^{n-1})
Different types of polynomials encompass various essential concepts and properties, each delineated in detail in the provided table.
Algebraic polynomial identity
(a + b)^{2} = a^{2} + b^{2} + 2ab
(a - b)^{2} = a^{2} + b^{2} - 2ab
(a + b)(a - b) = a^{2} - b^{2}
(x + a)(x + b) = x^{2} + (a + b)x + ab
(x + a)(x - b) = x^{2} + (a - b)x - ab
(x - a)(x + b) = x^{2} + (b - a)x - ab
(x - a)(x - b) = x^{2} - (a + b)x + ab
(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)
(a - b)^{3} = a^{3} - b^{3} - 3ab(a - b)
(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2xz
(x + y - z)^{2} = x^{2} + y^{2} + z^{2} + 2xy - 2yz - 2xz
(x -y + z)^{2} = x^{2} + y^{2} + z^{2} - 2xy - 2yz + 2xz
(x - y - z)^{2} = x^{2} + y^{2} + z^{2} - 2xy + 2yz - 2xz
x^{3} + y^{3} + z^{3} - 3xyz = (x + y + z)(x^{2} + y^{2} + z^{2} - xy - yz - xz)
x^{2} + y^{2} = 12[(x + y)^{2} + (x - y)^{2}]
(x + a)(x + b)(x + c) = x^{3} + (a + b + c)x^{2} + (ab + bc + ca)x + abc
x^{3} + y^{3} = (x + y)(x^{2} - xy + y^{2})
x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2})
x^{2} + y^{2} + z^{2} - xy - yz - zx = 1/2[(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]
The division algorithm for polynomials states that when considering any two polynomials p(x) and g(x). Where g(x) is not equal to zero, Then we can find polynomials q(x) and r(x) such that:
p(x) = q(x) g(x) + r(x)
Here, r(x) = 0 or deg r(x) ≥ deg g(x)
Free download NCERT Solutions for Class 10 Maths Chapter 2 Polynomials PDF for CBSE Exam.
NCERT Solutions for Chapter 2 Maths Class 10 Polynomials Excercise: 2.1
Class 10 Maths Chapter 2 Solutions Polynomials Excercise: 2.2
Answer:
x ^{ 2 } - 2x - 8 = 0
x ^{ 2 } - 4x + 2x - 8 = 0
x(x-4) +2(x-4) = 0
(x+2)(x-4) = 0
The zeroes of the given quadratic polynomial are -2 and 4
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
The zeroes of the given quadratic polynomial are 1/2 and 1/2
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
6x ^{ 2 } - 3 - 7x = 0
6x ^{ 2 } - 7x - 3 = 0
6x ^{ 2 } - 9x + 2x - 3 = 0
3x(2x - 3) + 1(2x - 3) = 0
(3x + 1)(2x - 3) = 0
The zeroes of the given quadratic polynomial are -1/3 and 3/2
Sum of roots:
Verified
Product of roots:
Verified
4u(u + 2) = 0
The zeroes of the given quadratic polynomial are 0 and -2
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
The zeroes of the given quadratic polynomial are and
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
3x ^{ 2 } + 3x - 4x - 4 = 0
3x(x + 1) - 4(x + 1) = 0
(3x - 4)(x + 1) = 0
The zeroes of the given quadratic polynomial are 4/3 and -1
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
Answer:
The required quadratic polynomial is
The required quadratic polynomial is
The required quadratic polynomial is x ^{ 2 } + .
The required quadratic polynomial is x ^{ 2 } - x + 1
The required quadratic polynomial is 4x ^{ 2 } + x + 1
The required quadratic polynomial is x ^{ 2 } - 4x + 1
NCERT Solutions for Chapter 2 Maths Class 10 Polynomials Excercise: 2.3
The quotient is x-3 and the remainder is 7x-9
The division is carried out as follows
The quotient is
and the remainder is 8
The quotient is and the remainder is
After dividing we got the remainder as zero. So is a factor of
After division, the remainder is zero thus is a factor of
The remainder is not zero, there for the first polynomial is not a factor of the second polynomial
Q3 Obtain all other zeroes of , if two of its zeroes are
Answer:
Two of the zeroes of the given polynomial are .
Therefore two of the factors of the given polynomial are and
is a factor of the given polynomial.
To find the other factors we divide the given polynomial with
The quotient we have obtained after performing the division is
(x+1) ^{ 2 } = 0
x = -1
The other two zeroes of the given polynomial are -1.
remainder =-2x+4
Carrying out the polynomial division as follows
Q5 (2) Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg q(x) = deg r(x)
example for the polynomial which satisfies the division algorithm with r(x)=0 is given below
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Excercise: 2.4
p(x) = 2x ^{ 3 } + x ^{ 2 } -5x + 2
p(1) = 2 x 1 ^{ 3 } + 1 ^{ 2 } - 5 x 1 + 2
p(1) =2 + 1 - 5 + 2
p(1) = 0
p(-2) = 2 x (-2) ^{ 3 } + (-2) ^{ 2 } - 5 x (-2) +2
p(-2) = -16 + 4 + 10 + 2
p(-2) = 0
Therefore the numbers given alongside the polynomial are its zeroes
Verification of relationship between the zeroes and the coefficients
Comparing the given polynomial with ax ^{ 3 } + bx ^{ 2 } + cx + d, we have
a = 2, b = 1, c = -5, d = 2
The roots are
Verified
Verified
Verified
p(x) = x ^{ 3 } - 4x ^{ 2 } + 5x - 2
p(2) = 2 ^{ 3 } - 4 x 2 ^{ 2 } + 5 x 2 - 2
p(2) = 8 - 16 + 10 - 2
p(-2) = 0
p(1) = 1 ^{ 3 } - 4 x 1 ^{ 2 } + 5 x 1 - 2
p(1) = 1 - 4 + 5 - 2
p(1) = 0
Therefore the numbers given alongside the polynomial are its zeroes
Verification of relationship between the zeroes and the coefficients
Comparing the given polynomial with ax ^{ 3 } + bx ^{ 2 } + cx + d, we have
a = 1, b = -4, c = 5, d = -2
The roots are
Verified
Verified
Verified
Hence the required cubic polynomial is x ^{ 3 } - 2x ^{ 2 } - 7x + 14 = 0
Q3 If the zeroes of the polynomial are a – b, a, a + b, find a and b.
The roots of the above polynomial are a, a - b and a + b
Sum of the roots of the given plynomial = 3
a + (a - b) + (a + b) = 3
3a = 3
a = 1
The roots are therefore 1, 1 - b and 1 + b
Product of the roots of the given polynomial = -1
1 x (1 - b) x (1 + b) = - 1
1 - b ^{ 2 } = -1
b ^{ 2 } - 2 = 0
Therefore a = 1 and .
Q4 If two zeroes of the polynomial are , find other zeroes .
therefore the factors are
We have to find the remaining two factors. To find the remaining two factors we have to divide the polynomial with the product of the above factors
Now carrying out the polynomial division
Now we get
So the zeroes are
Given the remainder =x+a
The obtained remainder after division is
now equating the coefficient of x
which gives the value of
now equating the constants
Therefore k=5 and a=-5
The degree of a polynomial
Number of zeroes in a quadratic polynomial
The sum & product of quadratic polynomial
Division algorithm
The number of zeroes in a cubic polynomial
Chapter No. | Chapter Name |
Chapter 1 | |
Chapter 2 | Polynomials |
Chapter 3 | |
Chapter 4 | |
Chapter 5 | |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | |
Chapter 13 | |
Chapter 14 | |
Chapter 15 |
The chapter 2 Maths Class 10 are important for board exams as well as for competitive exams.
These NCERT solutions are prepared by experts. These solutions are very well explained and are to the point.
To make the Class 10 Maths chapter 2 solutions easy to understand, relevant diagrams or images are given with the solutions wherever required.
You can check these solutions online or can save the webpage to get these solutions offline. NCERT Solutions for Class 10 Maths Chapter 2 PDF download will also be made available soon.
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You will find yourself much improved in terms of concepts, their applications, and problem-solving. 90% of the questions in the board examinations come from the NCERT syllabus.
Now when you have gone through the NCERT solutions for Maths chapter 2 Polynomials Class 10 , you must have learnt to answer a question in the step by step manner.
Once you have studied the Class 10 Maths chapter 2 NCERT solutions, your next target should be previous year questions papers.
NCERT syllabus coverage and previous year papers are enough tools to get a good score in the board examination. After going through these two things, you can jump to the next chapters.
At the careers360 official website, you can find polynomials class 10 ncert solutions. These solutions for ch2 maths class 10 are designed and developed by reliable experts keeping in mind the new pattern provided by the state and central boards. Also, you can download it as a polynomials class 10 pdf.
The topics covered in NCERT maths class 10 chapter 2 (Polynomials) are:
Definition and types of polynomials
Zeroes of a polynomial
Division algorithm for polynomials
Fundamental Theorem of Algebra
Relationship between zeroes and coefficients of a polynomial
Nature of roots of a polynomial
There are several reasons why you should use the ncert solutions for class 10 maths chapter 2 polynomials:
The NCERT Solutions are accurate and reliable
The NCERT Solutions cover all the concepts and exercises given in the textbook
The NCERT Solutions can help you develop problem-solving skills
The NCERT Solutions can be used as a reference guide
Overall, using the polynomials class 10 questions with answers can be a useful resource for students to improve their understanding of the concepts and perform well in exams.
There are a total of 10 exercises in Chapter 2 (Polynomials) of the NCERT Solutions for Class 10 Mathematics. These exercises consist of a variety of questions, including multiple choice questions, fill in the blanks, true or false, and short answer questions. The number of questions in each exercise may vary.
According to the NCERT chapter 2 Class 10 Maths Polynomials, a polynomial is a mathematical expression that consists of variables (such as x, y, etc.) and coefficients (numbers), which are combined using only the operations of addition, subtraction, and multiplication.
Application Date:20 November,2023 - 19 December,2023
Application Date:20 November,2023 - 19 December,2023
The eligibility age criteria for class 10th CBSE is 14 years of age. Since your son will be 15 years of age in 2024, he will be eligible to give the exam.
According to CBSE norms, a student must be 14 years old by the end of the year in which the exam will be held in order to sit for the 10th board exam. If your age is greater than 14, however, there are no limits. Therefore, based on your DOB, you will be 12 years and 6 months old by the end of December 2022. After the actual 16 June 2024, you'll be qualified to take your 10th board.
Hello aspirant,
Central Board Of Secondary Education(CBSE) is likely to declare class 10 and 12 terms 2 board result 2022 by July 15. The evaluation process is underway. Students are demanding good results and don't want to lack behind. They are requesting the board to use their best scores in Term 1 and Term 2 exams to prepare for the results.
CBSE concluded board exams 2022 for 10, and 12 on June 15 and May 24. Exams for both classes began on April 26. A total of 35 lakh students including 21 lakh class 10 students and 14 lakh class 12 students appeared in exams and are awaiting their results.
You can look for your results on websites- cbse.gov (//cbse.gov) .in, cbseresults.nic.in
Thank you
Hello SIR CBSE board 9th class admission 2022 Kara Raha hu entrance ki problem hai please show the Entrance for Baal vidya mandir school sambhal
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Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Fashion journalism involves performing research and writing about the most recent fashion trends. Journalists obtain this knowledge by collaborating with stylists, conducting interviews with fashion designers, and attending fashion shows, photoshoots, and conferences. A fashion Journalist job is to write copy for trade and advertisement journals, fashion magazines, newspapers, and online fashion forums about style and fashion.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.
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Quality Assurance Manager Job Description: A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An IT Consultant is a professional who is also known as a technology consultant. He or she is required to provide consultation to industrial and commercial clients to resolve business and IT problems and acquire optimum growth. An IT consultant can find work by signing up with an IT consultancy firm, or he or she can work on their own as independent contractors and select the projects he or she wants to work on.
A Data Architect role involves formulating the organisational data strategy. It involves data quality, flow of data within the organisation and security of data. The vision of Data Architect provides support to convert business requirements into technical requirements.
The Security Engineer is responsible for overseeing and controlling the various aspects of a company's computer security. Individuals in the security engineer jobs include developing and implementing policies and procedures to protect the data and information of the organisation. In this article, we will discuss the security engineer job description, security engineer skills, security engineer course, and security engineer career path.
A UX Architect is someone who influences the design processes and its outcomes. He or she possesses a solid understanding of user research, information architecture, interaction design and content strategy.
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