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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) â‰Ą 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 â€“ 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.
Dear aspirant !
Hope you are doing well ! The class 10 Hindi mp board sample paper can be found on the given link below . Set your time and give your best in the sample paper it will lead to great results ,many students are already doing so .
Hope you get it !
Thanking you
Hello aspirant,
The dates of the CBSE Class 10th and Class 12 exams are February 15â€“March 13, 2024 and February 15â€“April 2, 2024, respectively. You may obtain the CBSE exam date sheet 2024 PDF from the official CBSE website, cbse.gov.in.
To get the complete datesheet, you can visit our website by clicking on the link given below.
https://school.careers360.com/boards/cbse/cbse-date-sheet
Thank you
Hope this information helps you.
Hello aspirant,
The paper of class 7th is not issued by respective boards so I can not find it on the board's website. You should definitely try to search for it from the website of your school and can also take advise of your seniors for the same.
You don't need to worry. The class 7th paper will be simple and made by your own school teachers.
Thank you
Hope it helps you.
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.
That totally depends on what you are aiming for. The replacement of marks of additional subjects and the main subject is not like you will get the marks of IT on your Hindi section. It runs like when you calculate your total percentage you have got, you can replace your lowest marks of the main subjects from the marks of the additional subject since CBSE schools goes for the best five marks for the calculation of final percentage of the students.
However, for the admission procedures in different schools after 10th, it depends on the schools to consider the percentage of main five subjects or the best five subjects to admit the student in their schools.
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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.
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.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
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 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.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
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.
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 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.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
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.
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
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.