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Polynomials Class 10 Notes CBSE Maths Chapter 2 (Free PDF Download)

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Exam-Focused Revision Notes for CBSE Class 10 Maths Chapter 2 - Polynomials

For young students, tenth grade plays a vital role in their entire schooling education. It will become the turning point for their further education. So, gaining knowledge is very important. Class 10 Maths Chapter 2 Notes professionally designed can be the perfect source of material to prepare for their exams. It has a downloading option from the official website. The material is prepared by well-experienced faculty with a full of practice questions.

Vedantu is a platform that provides free NCERT Book Solutions and other study materials for students. You can download the NCERT Solution for Class 10 Science to score more marks in the examinations.


Overview of Deleted Syllabus for CBSE Class 10 Maths Chapter 1 Real Numbers

Chapter

Dropped Topics

Polynomials

Page Number 33 - 37

2.4 Division algorithm for polynomials


Download CBSE Class 10 Maths Revision Notes 2024-25 PDF

Also, check CBSE Class 10 Maths revision notes for All chapters:


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Access Class 10 Maths Chapter 2 Polynomials

  • If \[\text{p}\left( \text{x} \right)\] is a polynomial in $\text{x}$, the degree of the polynomial \[\text{p}\left( \text{x} \right)\] is the largest power of $\text{x}$ in \[\text{p}\left( \text{x} \right)\].

  • Types of Polynomials:

  1. A linear polynomial is a polynomial with degree one.

  2. A quadratic polynomial is a polynomial with degree two.

  3. A cubic polynomial is a polynomial with degree three.

  • Zeros of a Polynomial:

If \[\text{p}\left( \text{x} \right)\] is a polynomial in $\text{x}$ and $\text{k}$ is any real number, the value obtained by substituting $\text{k}$ for $\text{x}$ in \[\text{p}\left( \text{x} \right)\] is known as the value of \[\text{p}\left( \text{x} \right)\] when \[\text{x = k}\]and is denoted by \[\text{p}\left( \text{k} \right)\]. If \[\text{p}\left( \text{k} \right)\text{ = 0}\], a real number $\text{k}$ is said to be a zero of a polynomial \[\text{p}\left( \text{x} \right)\].

  • The Geometrical Meaning of Polynomial Zeros:


The Geometrical Meaning of Polynomial Zeros


The Geometrical Meaning of Polynomial Zeros


  • The equation \[\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c}\] can have three cases for the graphs 

  1. Case (i): 

Here, the graph cuts \[\text{x-}\]axis at two distinct points \[\text{A}\]and \[\text{A }\!\!'\!\!\text{ }\].


The graph cuts \[\text{x-}\]axis at two distinct points \[\text{A}\]and \[\text{A }\!\!'\!\!\text{ }\]


The graph cuts \[\text{x-}\]axis at two distinct points \[\text{A}\]and \[\text{A }\!\!'\!\!\text{ }\]


  1. Case (ii): Here, the graph cuts the \[\text{x-}\]axis at exactly one point.


The graph cuts the \[\text{x-}\]axis at exactly one point


The graph cuts the \[\text{x-}\]axis at exactly one point


  1. Case (iii): Here, the graph is either completely above the \[\text{x-}\]axis or completely below the \[\text{x-}\]axis.


the graph is either completely above the \[\text{x-}\]axis or completely below the \[\text{x-}\]axis


the graph is either completely above the \[\text{x-}\]axis or completely below the \[\text{x-}\]axis


  • If \[\text{ }\!\!\alpha\!\!\text{ }\] and \[\text{ }\!\!\beta\!\!\text{ }\] are the zeroes of the quadratic polynomial\[\text{p}\left( \text{x} \right)\text{ = a}{{\text{x}}^{\text{2 }}}\text{+ bx + c, a}\ne \text{0}\], then it is known that \[\text{x -  }\!\!\alpha\!\!\text{ }\] and \[\text{x -  }\!\!\beta\!\!\text{ }\] are the factors of \[\text{p}\left( \text{x} \right)\].

  1. \[\text{ }\!\!A\!\!\text{  +  }\!\!\beta\!\!\text{  = - }\dfrac{\text{b}}{\text{a}}\]

  2. \[\text{ }\!\!\alpha\!\!\text{  }\!\!\beta\!\!\text{  = }\dfrac{\text{c}}{\text{a}}\]

  • Division Algorithm for Polynomials: 

  • If \[\text{p}\left( \text{x} \right)\] and \[\text{g}\left( \text{x} \right)\] are any two polynomials with \[\text{g}\left( \text{x} \right)\ne 0\], then polynomials \[\text{q}\left( \text{x} \right)\] and \[\text{r}\left( \text{x} \right)\] can be found such that  \[\text{p}\left( \text{x} \right)\text{= g}\left( \text{x} \right)\text{  }\!\!\times\!\!\text{  q}\left( \text{x} \right)\text{ + r(x)}\], where \[\text{r}\left( \text{x} \right)=0\] or degree of \[\text{r}\left( \text{x} \right)<\] degree of \[\text{g}\left( \text{x} \right)\].

  • This result is known as the Division Algorithm for polynomials.  

  • An example would make it easier to understand. So, consider a cubic polynomial x3 - 3x2 - x + 3.

  • Assuming that one of its zeroes is $1$, it is clear that \[\text{x - 1}\] is a factor of x3 - 3x2 - x + 3.

  • So, x3 - 3x2 - x + 3 can be divided by x -1 Taking out this factor, (x - 1)(x2 - 2x - 3).

  • Next, get the factors of x2 - 2x - 3 by splitting the middle term. (x + 1)(x - 3).

x3−3x2−x+3=(x−1)(x+1)(x−3)

  • So, all the three zeroes of the cubic polynomial are  $1, − 1, 3$.


Mastering Polynomials: Comprehensive Class 10 Notes for Polynomial - A Chapter Overview

Class 10 Maths Chapter 2 Notes is available in a PDF format on the main website of Vedantu. Students can avail of this opportunity to take a physical copy, which avoids internet issues. This pdf is also helpful to store for the future. It is also useful to refer during the time of entrance test, olympiads, and other examinations. 


Introduction:- 

The Notes of ch 2 Maths Class 10 has started the lesson similar to all the chapters with an introduction. In this introduction part, the notes have provided a quick recap of all the previous knowledge students have gained in the earlier classes. A polynomial is an equation with one variable and changes its type based on the variable's degree. If the variable has a single degree, then it is called the linear polynomial. If the variable's degree is 2, then the polynomial is set to be quadratic polynomial. Similarly, the cubic polynomial is the polynomial that has a degree of 3. It is also explained that the value of the polynomial and if the value is 0, then it is called 0 polynomial.


CBSE Class 10 Math Revision Notes


Geometrical Meaning of The Zeros of A Polynomial

As it is already clear that the zero polynomial is a polynomial whose value is zero. In addition to the ordinary polynomials, the zero polynomial is very important when compared to the other. Because while plotting a graph for the zero quadratic polynomial, we have a result of the parabola. This parabola also appears at different points, which represent different cases. Those cases are specified by Notes of Chapter Polynomials Class 10. 

  • One parabola may cut the x-axis and y-axis at two points.

  • When one parabola may cut the x-axis and y-axis at a single point.

  • The parabola may be entirely above the x-axis or completely below the x-axis.

  • One parabola may be completely above the y-axis or completely below the y-axis.

These are the various cases observed in the zero polynomial while plotting graphs are explained with several solved examples by taking different values.


Relationship Between Zeros And Coefficients of a Polynomial

In this section, the Notes of Chapter 2 Maths Class 10 explains the questions of a polynomial by reminding the factorization method, which had been learned in the previous class. In the factorization, the quadratic polynomial has been split into multiple factors, which means writing each term separately. Then taking a common term and obtaining the result. The remaining two terms can be treated as factors.

For an instance,

3x2+14x+ 8 is the polynomial.

Then split the equation into four terms without changing the values.

3x2+ 2x +12x +8 = 0

Then take common and split into two factors.

(3x+2)(x+4)=0

Hence the factors are, -⅔, -4.


Division Algorithm For Polynomials

After understanding the questions and factors, the Class 10 Maths ch 2 Notes notes the division algorithm concerning polynomials. So far, the PDF has discussed quadratic polynomials. But to understand the division algorithm, Class 10 Maths Polynomials Notes specifies that we need to consider cubic polynomials with three zeros. By taking a single value, we need to learn the finding process of the other two values.

  • To get the first term of coefficient, we need to divide the highest degree term of dividend.

  • The same process has to be followed for the next term also.

  • If the degree of the divisor is less than the degree of dividend, then we need to use a formula to find the quotient.

Dividend = divisor * quotient + Remainder. 

This is the process and formula for division algorithm explained by Chapter 2 Maths Class 10 Notes


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Conclusion

The Class 10th Maths Chapter 2 Notes wrapped off the chapter's last portion by reiterating many key ideas and polynomials' rules in the form of pointers. Exercises and solution tests were also included in the pdf for each subject. The value of the polynomial and the degree of the polynomial were highly essential and kept changing, according to the Class 10 Chapter 2 Maths Notes. They have a variety of cases at various values.

FAQs on Polynomials Class 10 Notes CBSE Maths Chapter 2 (Free PDF Download)

1. Find the value of “p” from the polynomial x2 + 3x + p, if one of the zeroes of the polynomial is 2.

It is given that, 2 is the zero of the polynomial.


We know that if α is a zero of the polynomial p(x), then p(α) = 0


So,Substituting x = 2 in x2 + 3x + p,


We get,  22 + 3(2) + p = 0

              4 + 6 + p = 0

              10 + p = 0

                p = -10.

Hence the value of p is -10.

2. How many zeros do the polynomial  (x – 3)2 – 4 can have? Also, find them.

Given the polynomial equation is (x – 3)2 – 4


Now, we need to expand this equation.


 x2 + 9 – 6x – 4


x2– 6x + 5


As the equation has a degree of 2, it is called a quadratic polynomial. So,  the number of zeroes will be 2.


Now, solve x2 - 6x + 5 = 0 to get the factors by using the  factorization method.


So, x2– x – 5x + 5 = 0


 x(x – 1) -5(x – 1) = 0


 (x – 1)(x – 5) = 0


x = 1, x = 5


So, the factors are 1 and 5.


These are the two zeros that we need to find.

Hence it is solved. 

3. What is polynomial in the context of Chapter 2 Class 10?

The term Polynomial comes from the word 'poly' which means 'many' and 'nominal' which means term. Therefore, polynomials refer to many terms. A polynomial is made up of many terms which can only be substrated, added, or multiplied. A polynomial with the highest exponent of variables is known as the degree of the polynomial. The polynomial with degree one is known as a linear, degree two is known as a quadratic, and degree three is known as a cubic polynomial. 

4. What are important polynomial notes for chapter 2 class 10?

While studying Chapter 2 Class 10 Maths, it is important for you to understand the concept of the polynomial. This is an important chapter from your board examination point of view. The important polynomial notes include types of polynomials, the degree of polynomials, zeros of a polynomial, formulas, and all the algorithms. These notes will help you during your revision time as well, to help you score well during your board exam. 

5. Where can I find NCERT Solutions for Class 10 Maths Chapter 2?

Vedantu provides students with the best NCERT Solutions for the Class 10 Maths Chapter 2. It provides all important questions and solutions for Chapter 2 “Polynomials''. It covers all types of questions from the board examination point of view. They have a variety of questions to help students understand the chapters and important concepts better. These solutions and questions are extremely important for all CBSE students from the viewpoint of examinations. Also, the solutions PDF is available for free download on the Vedantu mobile app.

6. How can you divide one polynomial with another polynomial?

A polynomial refers to an expression with more than two algebraic terms. It refers to the sum of many terms that have different powers of the same variable. You can divide one polynomial with another polynomial using two methods namely long division, and synthetic division. The easiest way to divide one polynomial with another polynomial is using long division. Using long division will help to test whether or not a single polynomial has another one as a factor.

7. How to ace Class  10 Maths Chapter 2?

Class 10 Maths Chapter 2 is an important topic from the examination point of view. Students can rely on the study materials available on the official website of  Vedantu or their app and they are available free of cost. One of the most important ways to ace Chapter 2 is to complete all the NCERT questions and answers and carry out thorough revision. Revision is an important part of exam preparation. By completing solutions and exercises the students can ace their Class 10 Maths Chapter 2.