NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.4

CBSE Class 10 NCERT Solutions for Maths Chapter 2 Polynomials Exercise 2.4

NCERT Chapter 2 Class 10 Maths Exercise 2.4 gives you an inclusive idea about Polynomials. From this chapter, you will know what Polynomials actually are, their relevance, and their importance. The NCERT Solutions for Class 10 Maths Chapter 2 will help you in understanding the concept of Polynomials deftly. The Polynomials Class 10 NCERT Chapter Solution is important for you to understand many significant concepts later in Maths. Download Exercise 2.4 Class 10 Maths NCERT Solutions PDF for free now. You can also Download NCERT Solutions for Class 10 Maths to help you to revise the complete Syllabus and score more marks in your examinations. Also, download NCERT Solutions Class 10 Science and score more in your science exam as well.


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Chapter 2-Polynomials part-1

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.4

Polynomials are nothing but a mathematical expression that comprises of coefficients or variables. Polynomials always consist of positive integer exponents. Sounds complicated? Don’t worry. Download Vedantu’s PDF on NCERT Chapter 2 Class 10 Maths on Polynomials Exercise 2.4 and get all your doubts cleared seamlessly. The solutions that are provided to you are very easily understandable. Try it yourself to improve your studying process. 

Clear the Concept First

Clearing the concepts of Chapter 2 Maths Class 10 that deals with Polynomials is essential for any student. This is because, without a proper understanding of what polynomials are, it is impossible to solve these problems.  It is essential to learn how to resolve a complicated set of numbers into a simple polynomial for any problem-solving. Vedantu’s notes on Ch 2 Maths Class 10 include all questions and their smart solutions that help any student learn how to solve the problems. Download the PDF On Polynomials Class 10 now.

Practice Problems

Vedantu’s notes on the Polynomial Class 10 Chapter comprise of all the probable questions that can be a part of your question paper in any examination. The solutions are also provided with several tricks to help students get a clear idea about the concepts. Now, no one can stop you from being excellent in Maths. 

Knowing About Polynomials Is Inevitable

If a student doesn’t get a precise idea of what polynomials are in 10th standard, it is sure to get very difficult for them to solve difficult math questions in future classes. Vedantu’s notes will effectively help you to prepare yourself for the next term. The Chapter that has been curated by the CBSE board is sure to give you all the knowledge about Polynomials and Vedantu will help you get a grip on the concept with very little effort. 

No Need to Put In Too Much Effort

Vedantu’s Notes on Chapter 2 Maths Class 10 makes the process of grasping the concept of polynomials very simple. A thorough practice of the solutions just once or twice will make all the difference. You can be sure of the fact that noting will be asked in the question paper outside the concepts and questions solved in the notes provided by Vedantu. The notes are for you and Vedantu knows what you need.


Facts

  • An expression of the form p(x) = a0 + a1x + a2x2 + a3x3 + ……. + anxn is called a polynomial, where a0, a1, a2, a3,……., an are real numbers, called coefficients of the polynomial, and n is non-negative integer.

Example: x + 2, 3x2 + √5x - √6 etc.

  • The highest power of the variable appearing in the polynomial is called its degree. For example,

  1. degree of x2 + 2x + 3 is 2.

  2. degree of √3x + 5 is 1

  • The degree of the zero polynomial is not established.

  • A linear polynomial is one degree of the polynomial. For example: x +1, 2x + 5, etc. Its general form is ax + b, where a, b and c are real numbers and a≠ 0.

  • A polynomial that is of a degree two is called a quadratic polynomial. 

For example: x2 + x + 2, 2x2 - 3x + 4, etc. Its general form is ax2 + bx + c, where a, b and c are real numbers and a ≠ 0.

  •  A polynomial that is of a degree 3 is called a cubic polynomial. 

For example: x3 + x2 + x + 1, 2x3 + x2 – 3x – 4, etc. Its general form is ax3 + bx2 + cx + d, where a, b, c and d are real numbers and a ≠ 0.

  •  A biquadratic polynomial is a polynomial of degree 4. 

For example: x4 + x3 + 5x2 + 6x + 2, etc. Its general form is ax4 + bx3 + cx2 + dx + e, where a, b, c, d and e are real numbers and a ≠ 0.

  • A real number is said to be a zero of a polynomial.  p(x), if p(k) = 0, where k is the real number.

For example: if p(x) = x2 – 4x + 4, and p(2) = (2)2 – 4(2) + 4 = 4 – 8 + 4 = 0, then 2 is a zero of polynomial p(x).

  • Remainder theorem: To understand the remainder theorem, we must have knowledge about factors and multiples,  long division algorithms. 

If f(x) is any polynomial of a degree 1 and f(x) is divided by the linear polynomial x – a, then the remainder is f(a).

  • Factor theorem: Sometimes a polynomial having values that are unknown and one of its factors are provided and we have to calculate the value of that unknown value. Sometimes, a linear polynomial is given and we have to verify whether it is a factor of given polynomial f(x)of degree greater than 1 or not. For solving such types of problems factor theorem is required. 

In this segment, we will study division algorithms for polynomials.

If any two polynomials like p(x) and g(x) along with one polynomial of the two is not equal to 0 i.e.,  g(x) ≠ 0, then we can ascertain the polynomials r(x) and q(x) such that p(x) = g(x) * q(x) + r(x) = 0 or degree of r(x) < degree of g(x). The outcome of the above numerical is known as the Division Algorithm for polynomials. 

In other words, we can say that when p(x) divided by g(x), we get q(x) as quotient and r(x) as remainder. 

Thus, Dividend = Divisor x Quotient + Remainder

Note: If r(x) = 0, then g(x) is a factor of f(x).


Method of Division Polynomials

  1. Organize the terms of the dividend and the divisor in descending order of their degrees.

  2. The first term of the dividend must be divided by the first term of the divisor to obtain the first term of the quotient.

  3. Multiply the whole divisor by the first term if the quotient and subtract the result obtained from the dividend.

  4. Observe the remainder as the new dividend and proceed as earlier.

  5. Repeat the procedure till a remainder is derived which is either 0 or whose degree is less than that of the divisor. 


Applications of Division algorithm

1. Verifying the division algorithm for polynomials

Let us consider the following example.

Let us divide  the polynomial f(x) = x3 - 3x2 + 2x - 5 by the polynomial g(x) = x2 -2x - 3 and verify the 

division algorithm.

    __________________________

x2 -2x - 3 )  x3 - 3x2 + 2x - 5     ( x-1

        x3 - 2x2 - 3x 

      ________________________

      - x2 + 5x - 5

      -  x2 + 2x +3

    __________________________

                  3x - 8

∴ Quotient: q(x) = x-1 and remainder r(x)  = 3x -8.

     Now, Quotient x Divisor + Remainder

          =  (x-1) (x2 - 2x - 3) + 3x - 8

          =  x3 -2x2 - 3x - x2 + 2x + 3 + 3x - 8

          =  x3 - 3x2 + 2x - 5 = Dividend

∴The division algorithm is verified.


2. To find the Quotient and Remainder using a division algorithm. Dividing f(x) = x3 - 6x2 + 11x - 5 and g(x) = x+ 1

Sol. Here, degree of f(x) = 3 & degree of g(x) = 1

∴ Degree of quotient g(x) = 3 -1 = 2

  & degree of remainder r(x) = 0

Let q(x) = ax2 + bx + c & r(x) = k 

By division algorithm, we have

f(x) = q(x) = g(x) + r(x)

=>         x3 - 6x2 + 11x - 5 = (ax2 + bx + c )(x + 1)  + k

=> x3 - 6x2 + 11x - 5 = ax3 + (a + b) x2 + (b + c)x + (c + k)

Equating the quotients of various powers of x on both sides, we get

1 = a (equating coefficients of x3)

-6 = a + b => b = -7 (equating coefficient of x2)

11 = b + c => c = 18 (equating coefficient of x1)

& -5 = c + k => k = -23 (equating constant terms)

∴ q(x) = ax2 + bx + c = x2 - 7x + 18

      & r(x) = -23.

3. To check whether a given polynomial is a factor of the other polynomial by applying a division algorithm. Checking whether g(x) = x2 - 2x + 1 is a factor of the polynomial f(x) = x4 -4x3 + 6x2 - 4x + 1.

Sol. Here degree of f(x) = 4 & degree of g(x) = 2

∴degree of quotient q(x) = 4 -2  & remainder r(x) will have degree 1 or less .

Let q(x) = ax + bx + c & r(x) = mx + n 

using division algorithm , we have f(x) = q(x)g(x) + r(x)

=> x4 - 4x3 + 6x2 - 4x + 1 = (ax2 + bx + c) (x2 -2x + 1) _ mx + n

=> x4 - 4x3 + 6x2 - 4x + 1 = ax4 + ( -2a + b) x3 + (a -2b + c) x2 + (b - 2c + m) x + (c + n)

Equating the coefficients of x4, x3, x2, x1 & constant terms on both sides of the equation 

respectively, we get

x4 → 1 = a

x3 → -4 = -2a + b = -2 1+ b => b = -2

x2 → 6 = a - 2b + c =1 - 2 (-2) + c  => c = 1

x → -4 = b - 2c + m = -2 -2 1 + m => m = 0

constant → 1 = c + n = 1 + n => n = 0

∴Quotient, q(x) = ax2 + bx + c = x2 -2x + 1 & remainder, r(x) = mx + n = 0x + 0  = 0

As, r = 0, g(x) is a factor of f(x).


4. To find the remaining zeroes of a polynomial when some of its zeroes are given. Consider the polynomial f(x) = 2x4 - 3x3 - 3x2 - 6x -2, if two of its zeroes are √2 & - √2, find the other zero.

Sol. If x = is a zero of a polynomial f(x), then x - is a factor of f(x)

        As, √2 & - √2 are zeroes of f(x)

        ∴(x - √2) & ( x + √2) are factors of f(x)

        ∴(x - √2)  ( x + √2), i.e x2 - 2 is a factor  of f(x)

        To find the other zeroes of f(x), we divide f(x) by g(x) = x2 - 2.

          ____________________________

x2 - 2 ) 2x4 - 3x3 - 3x2 - 6x -2    ( 2x2 - 3x + 1

2x4- 4x2

_________________________________

    - 3x3 + x2 - 6x -2 

  - 3x3          + 6x

________________________

  x2       -2

                 x2        -2

___________________________

      0

By division algorithm, we have

2x4 - 3x3 - 3x2 - 6x -2  = (x2 - 2) (2x2 - 3x + 1)

= (x2 - 2) (2x2 - 2x - x + 1)

= (x2 - 2) {2x(x-1) - (x - 1)}

= (x - √2) (x + √2) (x - 1) (2x -1)

Hence, the zeroes of the given polynomial apart from √2 & - √2 are 1 and ½ .


Why Vedantu?

Vedantu makes Mathematics interesting for even those students who are scared even by the name of the subject. Highly qualified and experienced teachers have curated the contents of the solution. Vedantu’s notes are the best if you want to get your concepts cleared easily at the primary stage.

You don’t need to go to Maths tuition every week once or twice after your hectic school hours anymore. All you need to do is open the website or the app of Vedantu and download the PDF on Polynomials Class 10 for free.

If you go through Vedantu’s notes and practice Maths religiously, you will get to see the difference in yourself in your next examination. You don’t need to practice Maths hours after hours anymore. All you need to do is to go through the questions and solutions provided by Vedantu. There is no important question on the Maths Chapter 2 Class 10 that Vedantu has missed in its notes.

With Vedantu’s Solutions, you will also learn how to solve a problem related to the Polynomial Chapter in the easiest manner possible. Once you learn it, no one can stop you from getting your dream marks in Maths. This time with Vedantu’s notes, you will be surely able to surprise your teachers, parents, and friends. Download the Exercise 2.4 Class 10 Maths NCERT Solutions for free to score better in your exams.

FAQs (Frequently Asked Questions)

1. What does NCERT Solutions Class 10th Maths Chapter 2 Exercise 2.4 deal with?

NCERT Chapter 2 Class 10 Maths Exercise 2.4 gives you a thorough idea about Polynomials. With the help of this chapter, you will be able to learn about the minute details of Polynomials, its definition and importance.


The NCERT Solutions for Class 10 Maths Chapter 2 will help you in understanding the concept of Polynomials in a neat way. This chapter holds sheer importance for a student as this helps to understand many other significant concepts in Maths. You can download


The Exercise 2.4 Class 10 Maths NCERT Solutions given in PDF format for free of cost. It will help you to revise the entire syllabus and score the highest possible marks in the exams.

2. What are the benefits of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4?

The benefits of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4 are many as these are given in the following. 

  • These solutions follow NCERT guidelines which help in preparing the students accordingly.

  • After going through the stepwise solutions to the problems from the exercise, which are solved by our subject matter experts, you will be able to score high in the exams.

  • It contains all the important questions from the Class 10 Maths Chapter 2- Polynomials.

  • It helps to score the highest marks in maths exams.

  • You can solve all the questions of exercise 2.4 efficiently and quickly revise all the concepts.

3. Where will I get the best quality NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4?

The best quality NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4 are available on the official website of Vedantu, the leading ed-tech portal in India. These solutions for Class 10 Maths Chapter 2 Polynomials are prepared by following NCERT guidelines so that it should cover the whole syllabus accordingly. These are very helpful in scoring well in examinations. Our in-house Maths experts provide a detailed, step-wise solution of each question given in the exercises 2.4 in the NCERT textbook for Class 10.

4. Can I download the NCERT Solutions for Class 10 Maths Chapter 2- Polynomials Exercise 2.4 from Vedantu app?

Yes, definitely. You can download the PDF files of NCERT solutions for class 10 maths chapter 2 – polynomials. Exercise 2.4 is the fourth exercise of Chapter 2 of class 10 Maths. Polynomials are one of the important topics in class 10th Maths syllabus and it was introduced in Class 9.

5. How can I download the NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths?

Students of Class 10 can download the NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths online. All NCERT Solutions are available in PDF format and it is easy to download. Students can download the NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths in this way:

  • Visit the NCERT solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths available free of cost on Vedantu’s website and app.

  • Click on download to download the solutions and save the file on your computer.

These solutions are free of cost and you can use the PDF file even when you are offline to study for your exams. 

6. What are the topics included in Class 10 Maths Chapter 2 Exercise 2.4?

Exercise 2.4 of Chapter 2 of Class 10 Maths covers the division algorithm of polynomials. Students can understand the main topic given in Exercise 2.4 by doing all NCERT questions. Students can refer to the NCERT Solutions for Exercise 2.4 on Vedantu. All Solutions are explained in detail for a clear understanding of the topic given in Exercise 2.4. Students can clear all doubts by doing all the solutions given on Vedantu. 

7. What are the benefits of studying from NCERT Solutions of Exercise 2.4 of Chapter 2 of Class 10 Maths given on Vedantu?

NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths are given in a stepwise manner that can help students to understand the questions easily. Students can also score high marks by understanding the concepts and clearing their doubts related to Exercise 2.4. All NCERT Solutions are based on the pattern and style of questions asked in the exams and students can solve all NCERT Solutions to score high marks in their exams. 

8. Are NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths sufficient to prepare for the exams?

NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths are sufficient for the students to prepare for the exams as they can solve all NCERT solutions. The solutions are prepared by expert subject teachers from the exam’s point of view. Students can understand the style of questions asked in the exams that can help them to solve all questions easily in the exams. All solutions are available in PDF format to download easily for exam preparation. 

9. What are the tips to score full marks in Exercise 2.4 of Chapter 2 of Class 10 Maths?

Students of Class 10 can score full marks in Exercise 2.4 of Chapter 2 of Class 10 Maths by practicing all questions given in the Exercise. Vedantu’s NCERT Solutions for Exercise 2.4 of Chapter 2 of Class 10 Maths can help you score good marks in Exercise 2.4. Students can practice all questions given in the Class 10 Maths Exercise 2.3 from Vedantu to understand the exam pattern and marking scheme. The solutions are prepared by expert subject teachers so that students can understand the concepts and score full marks in Exercise 2.4 of Chapter 2 of Class 10 Maths. 

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