• NCERT Solutions
    • NCERT Library
  • RD Sharma
    • RD Sharma Class 12 Solutions
    • RD Sharma Class 11 Solutions Free PDF Download
    • RD Sharma Class 10 Solutions
    • RD Sharma Class 9 Solutions
    • RD Sharma Class 8 Solutions
    • RD Sharma Class 7 Solutions
    • RD Sharma Class 6 Solutions
  • Class 12
    • Class 12 Science
      • NCERT Solutions for Class 12 Maths
      • NCERT Solutions for Class 12 Physics
      • NCERT Solutions for Class 12 Chemistry
      • NCERT Solutions for Class 12 Biology
      • NCERT Solutions for Class 12 Economics
      • NCERT Solutions for Class 12 Computer Science (Python)
      • NCERT Solutions for Class 12 Computer Science (C++)
      • NCERT Solutions for Class 12 English
      • NCERT Solutions for Class 12 Hindi
    • Class 12 Commerce
      • NCERT Solutions for Class 12 Maths
      • NCERT Solutions for Class 12 Business Studies
      • NCERT Solutions for Class 12 Accountancy
      • NCERT Solutions for Class 12 Micro Economics
      • NCERT Solutions for Class 12 Macro Economics
      • NCERT Solutions for Class 12 Entrepreneurship
    • Class 12 Humanities
      • NCERT Solutions for Class 12 History
      • NCERT Solutions for Class 12 Political Science
      • NCERT Solutions for Class 12 Economics
      • NCERT Solutions for Class 12 Sociology
      • NCERT Solutions for Class 12 Psychology
  • Class 11
    • Class 11 Science
      • NCERT Solutions for Class 11 Maths
      • NCERT Solutions for Class 11 Physics
      • NCERT Solutions for Class 11 Chemistry
      • NCERT Solutions for Class 11 Biology
      • NCERT Solutions for Class 11 Economics
      • NCERT Solutions for Class 11 Computer Science (Python)
      • NCERT Solutions for Class 11 English
      • NCERT Solutions for Class 11 Hindi
    • Class 11 Commerce
      • NCERT Solutions for Class 11 Maths
      • NCERT Solutions for Class 11 Business Studies
      • NCERT Solutions for Class 11 Accountancy
      • NCERT Solutions for Class 11 Economics
      • NCERT Solutions for Class 11 Entrepreneurship
    • Class 11 Humanities
      • NCERT Solutions for Class 11 Psychology
      • NCERT Solutions for Class 11 Political Science
      • NCERT Solutions for Class 11 Economics
      • NCERT Solutions for Class 11 Indian Economic Development
  • Class 10
    • NCERT Solutions for Class 10 Maths
    • NCERT Solutions for Class 10 Science
    • NCERT Solutions for Class 10 Social Science
    • NCERT Solutions for Class 10 English
    • NCERT Solutions For Class 10 Hindi Sanchayan
    • NCERT Solutions For Class 10 Hindi Sparsh
    • NCERT Solutions For Class 10 Hindi Kshitiz
    • NCERT Solutions For Class 10 Hindi Kritika
    • NCERT Solutions for Class 10 Sanskrit
    • NCERT Solutions for Class 10 Foundation of Information Technology
  • Class 9
    • NCERT Solutions for Class 9 Maths
    • NCERT Solutions for Class 9 Science
    • NCERT Solutions for Class 9 Social Science
    • NCERT Solutions for Class 9 English
    • NCERT Solutions for Class 9 Hindi
    • NCERT Solutions for Class 9 Sanskrit
    • NCERT Solutions for Class 9 Foundation of IT
  • CBSE Sample Papers
    • Previous Year Question Papers
    • CBSE Topper Answer Sheet
    • CBSE Sample Papers for Class 12
    • CBSE Sample Papers for Class 11
    • CBSE Sample Papers for Class 10
    • Solved CBSE Sample Papers for Class 9 with Solutions 2024-2025
    • CBSE Sample Papers Class 8
    • CBSE Sample Papers Class 7
    • CBSE Sample Papers Class 6
  • Textbook Solutions
    • Lakhmir Singh
    • Lakhmir Singh Class 10 Physics
    • Lakhmir Singh Class 10 Chemistry
    • Lakhmir Singh Class 10 Biology
    • Lakhmir Singh Class 9 Physics
    • Lakhmir Singh Class 9 Chemistry
    • PS Verma and VK Agarwal Biology Class 9 Solutions
    • Lakhmir Singh Science Class 8 Solutions

Learn CBSE

NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

Polynomials Class 9 Notes Maths Chapter 2

April 12, 2019 by Sastry CBSE

CBSE Class 9 Maths Notes Chapter 2 Polynomials Pdf free download is part of Class 9 Maths Notes for Quick Revision. Here we have given NCERT Class 9 Maths Notes Chapter 2 Polynomials.

CBSE Class 9 Maths Notes Chapter 2 Polynomials

1. Polynomial: A polynomial in one variable x is an algebraic expression of the form
Polynomials Class 9 Notes Maths Chapter 2 1

2. Terms: The several parts of a polynomial separated by ‘+’ or ‘-‘ operations are called the terms of the expression.

3. Types of Polynomial
(i) Constant polynomial: A polynomial containing one term only, consisting of a constant is called a constant polynomial.
e.g., -6, 4, \(\frac { 2 }{ 3 }\), \(\frac { -3 }{ 4 }\) etc., are still constant polynomial.
Generally, each real number is a constant polynomial.

(ii) Zero polynomial: A polynomial consisting of one term, namely zero only, is called a zero polynomial.

(iii) Monomial: Polynomials having only one term are called monomials (‘mono’ means ‘one’).
e.g., µ43, \(\frac { 7 }{ 3 }\) xz and – 2 are all monomials.

(iv) Binomial: Polynomials having only two terms are called binomials (‘bi’ means ‘two’).
e.g., (x2 + x), (y30 + √2) and (5x2y + 6xz) are all binomials.

(v) Trinomial: Polynomials having only three terms are called trinomials (‘tri’ means ‘three’).
e.g., (x4 + x3 + √2), (µ43 + µ7 + µ) and (8y – 5xy + 9xy2) are all trinomials.

4. The degree of a Polynomial: Highest power of the variable in a polynomial is the degree of the polynomial.

(a) In one variable: In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial.
e.g.,
√2 – y3 + y5 + 2y6 is a polynomial in y of degree 6.
6x + √3 is a polynomial in x of degree 1.

(b) In two or more variables: In case of a polynomial in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial.
e.g.,
3x3 – 7x2y2 + 8 is a polynomial in x and y of degree 4.
8a8b – 4ab + √2 is a polynomial in a and b of degree 9.

Note:
The degree of a non-zero constant polynomial is ‘zero’.
The degree of a zero polynomial is not defined.

5. Linear Polynomial: A polynomial of degree one is called a linear polynomial.
e.g.,
x + √7 is a linear polynomial in x, y and z.
√2 µ + 3 is a linear polynomial in µ.

6. Quadratic Polynomial: A polynomial of degree two is called a quadratic polynomial.
e.g;
xy + yz + zx is a quadratic polynomial in x, y and z.
x2 + 9x – \(\frac { 3 }{ 2 }\) is a quadratic polynomial in x.

7. Cubic Polynomial: A polynomial of degree three is called a cubic polynomial.
e.g.,
ax3 + bx2 + cx + d is a cubic polynomial in x and a, b, c, d are constants.
2y3 + 3 is a cubic polynomial in y.
9x2y + xy – 4 is a cubic polynomial in x and y.

8. Value of a Polynomial: Value of a polynomial p(x) at x = a is p(a).
e.g., If p(x) = x2 + 2x + 6 then, at x = 2, p(2) = 22 + 2 × 2 + 6 = 14

9. Zeroes of a Polynomial: Zeroes of a polynomial p(x) is a number a such that p(a) = 0.

  • Zero may be a zero of a polynomial.
  • Every linear polynomial has one and only one zero.
  • Zero of a polynomial is also called the root of the polynomial.
  • A non-zero constant polynomial has no zero.
  • Every real number is a zero of the zero polynomial.
  • A polynomial can have more than one zero.

The maximum number of zeroes of a polynomial is equal to its degree.

10. Remainder Theorem: Let p (x) be any polynomial of degree n greater than or equal to one (n ≥ 1) and let a be any real number. If p (x) is divided by the linear polynomial x – a, then the remainder is p (a).
Dividend = (Divisor × Quotient) + Remainder

11. Factor Theorem: Let q(x) be a polynomial of degree n ≥ 1 and a be any real number, then
(i) (x – a) is a factor of q (x), if q(a) = 0 and
(ii) q(a) = 0, if x – a is a factor of q (x).

12. The factorisation of Quadratic Polynomials
(i) By splitting the middle term: Suppose a quadratic polynomial is x2 + lx + m where l and m are constants.
Now, we split the middle term lx as ax + bx, so that ab = m. Then,
x2 + lx + m = x2 + ax + bx + ab = x(x + a) + b(x + a) = (x + a) (x + b)

(ii) By using factor theorem: Suppose, ax2 + bx + c be a quadratic polynomial. Let two factors be (x – α) and (x – β)
∴ a (x – α) (x – β) = ax2 + bx + c
⇒ ax2 – a(α + β)x + aαβ = ax2 + bx + c
On equating the coefficient of x and constant term.
We get, α + β = \(\frac { -b }{ a }\) and αβ = \(\frac { c }{ a }\)
On simplifying, we get the values of α and β.

(iii) The factorisation of cubic polynomials: The splitting middle term method is not applicable for cubic polynomials. Its need to find at least one factor first, and then adjust the given polynomial such that it becomes the product of term in which one of them is that factor and the other is a quadratic polynomial. Now, solve this quadratic polynomial by the previous method.

13. Algebraic Identities: An algebraic identity is an algebraic equation that is true for all values of the variable occurring in it.
Some algebraic identities are given below

  • (x + y)2 = x2 + 2xy + y2
  • (x – y)2 = x2 – 2xy + y2
  • x2 – y2 = (x + y)(x – y)
  • (x + a)(x + b)= x2 + (a + b)x + ab
  • (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
  • (x + y)3 = x3 + y3 + 3xy(x + y)
  • (x – y)3 = x3 – y3 – 3xy(x – y)
  • x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)
    If x + y + z = 0, then x3 + y3 + z3 = 3xyz

NCERT Notes for Class 9 Maths

  • Chapter 1 Number Systems Class 9 Notes
  • Chapter 2 Polynomials Class 9 Notes
  • Chapter 3 Introduction to Euclid’s Geometry Class 9 Notes
  • Chapter 4 Lines and Angles Class 9 Notes
  • Chapter 5 Triangles Class 9 Notes
  • Chapter 6 Coordinate Geometry Class 9 Notes
  • Chapter 7 Heron’s Formula Class 9 Notes
  • Chapter 8 Linear Equations in Two Variables Class 9 Notes
  • Chapter 9 Quadrilaterals Class 9 Notes
  • Chapter 10 Areas of Parallelograms and Triangles Class 9 Notes
  • Chapter 11 Circles Class 9 Notes
  • Chapter 12 Constructions Class 9 Notes
  • Chapter 13 Surface Areas and Volumes Class 9 Notes
  • Chapter 14 Statistics Class 9 Notes
  • Chapter 15 Probability Class 9 Notes

We hope the given CBSE Class 9 Maths Notes Chapter 2 Polynomials Pdf free download will help you. If you have any query regarding NCERT Class 9 Maths Notes Chapter 2 Polynomials, drop a comment below and we will get back to you at the earliest.

Filed Under: CBSE

LearnCBSE.in Student Education Loan
  • Student Nutrition - How Does This Effect Studies
  • Words by Length
  • NEET MCQ
  • Factoring Calculator
  • Rational Numbers
  • CGPA Calculator
  • TOP Universities in India
  • TOP Engineering Colleges in India
  • TOP Pharmacy Colleges in India
  • Coding for Kids
  • Math Riddles for Kids with Answers
  • General Knowledge for Kids
  • General Knowledge
  • Scholarships for Students
  • NSP - National Scholarip Portal
  • Class 12 Maths NCERT Solutions
  • Class 11 Maths NCERT Solutions
  • NCERT Solutions for Class 10 Maths
  • NCERT Solutions for Class 9 Maths
  • NCERT Solutions for Class 8 Maths
  • NCERT Solutions for Class 7 Maths
  • NCERT Solutions for Class 6 Maths
  • NCERT Solutions for Class 6 Science
  • NCERT Solutions for Class 7 Science
  • NCERT Solutions for Class 8 Science
  • NCERT Solutions for Class 9 Science
  • NCERT Solutions for Class 10 Science
  • NCERT Solutions for Class 11 Physics
  • NCERT Solutions for Class 11 Chemistry
  • NCERT Solutions for Class 12 Physics
  • NCERT Solutions for Class 12 Chemistry
  • NCERT Solutions for Class 10 Science Chapter 1
  • NCERT Solutions for Class 10 Science Chapter 2
  • Metals and Nonmetals Class 10
  • carbon and its compounds class 10
  • Periodic Classification of Elements Class 10
  • Life Process Class 10
  • NCERT Solutions for Class 10 Science Chapter 7
  • NCERT Solutions for Class 10 Science Chapter 8
  • NCERT Solutions for Class 10 Science Chapter 9
  • NCERT Solutions for Class 10 Science Chapter 10
  • NCERT Solutions for Class 10 Science Chapter 11
  • NCERT Solutions for Class 10 Science Chapter 12
  • NCERT Solutions for Class 10 Science Chapter 13
  • NCERT Solutions for Class 10 Science Chapter 14
  • NCERT Solutions for Class 10 Science Chapter 15
  • NCERT Solutions for Class 10 Science Chapter 16

Free Resources

RD Sharma Class 12 Solutions RD Sharma Class 11
RD Sharma Class 10 RD Sharma Class 9
RD Sharma Class 8 RD Sharma Class 7
CBSE Previous Year Question Papers Class 12 CBSE Previous Year Question Papers Class 10
NCERT Books Maths Formulas
CBSE Sample Papers Vedic Maths
NCERT Library

NCERT Solutions

NCERT Solutions for Class 10
NCERT Solutions for Class 9
NCERT Solutions for Class 8
NCERT Solutions for Class 7
NCERT Solutions for Class 6
NCERT Solutions for Class 5
NCERT Solutions for Class 4
NCERT Solutions for Class 3
NCERT Solutions for Class 2
NCERT Solutions for Class 1

Quick Resources

English Grammar Hindi Grammar
Textbook Solutions Maths NCERT Solutions
Science NCERT Solutions Social Science NCERT Solutions
English Solutions Hindi NCERT Solutions
NCERT Exemplar Problems Engineering Entrance Exams
Like us on Facebook Follow us on Twitter
Watch Youtube Videos NCERT Solutions App