Polynomial Class 9 Notes – Chapter 2

CBSE Class 9 Maths Polynomials Notes:-

Polynomial is derived from the words ‘poly’, which means ‘many’, and the word ‘nomial’, which means ‘term’. In maths, a polynomial expression consists of variables which are also known as indeterminates and coefficients. The coefficients involve the operations of subtraction, addition, non-negative integer exponents of variables and multiplication. In mathematics, both algebraic expressions and polynomials are made up of variables and constants, including arithmetic operations. The only difference between them is that algebraic expressions contain irrational numbers in the powers. A detailed polynomials Class 9 notes are provided here along with some important questions so that students can understand the concept easily.

Polynomials Class 9 Notes

To prepare for Class 9 exams, students will require notes to study. These notes are of great help when they have to revise chapter 2 polynomials before the exam. The notes provide a brief of the chapter so that students find it easy to have a glance at once. Go through the key points given and solve problems based on them.
The topics and subtopics covered in Class 9 polynomials chapter 2 include:

• Introduction
• Polynomials in One Variable
• Zeros of Polynomials
• Remainder Theorem
• Factorisation of Polynomials
• Algebraic Identities

Polynomial Definition

Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. An algebraic expression p(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + … a_n is a polynomial where a₀, a₁, ………. a_n are real numbers and n is non-negative integer.
Examples of polynomials are:

• 20
• x + y
• 7a + b + 8
• w + x + y + z
• x² + x + 1

Term

In the polynomial, each expression is called a term.  
Suppose x² + 5x + 2 is polynomial, then the expressions x², 5x, and 2 are the terms of the polynomial.  

Coefficient

Each term of the polynomial has a coefficient.
For example, if 2x + 1 is the polynomial, then the coefficient of x is 2.

Types of Polynomial

A polynomial of 1 term is called a monomial. Example: 2x.
A polynomial of 2 terms is called a binomial. Example: 5x + 2.
A polynomial of 3 terms is called a trinomial. Example: 2x + 5y – 4.

Constant Polynomial

Real numbers can also be expressed as polynomials. 3, 6, and 7 are also polynomials without any variables. These are called constant polynomials.  The constant polynomial 0 is called zero polynomial. The exponent of the polynomial should be a whole number. For example,  x^-2 + 5x + 2, cannot be considered a polynomial since the exponent of x is -2, which is not a whole number.

Degree of a Polynomial

The highest power of the polynomial is called the degree of the polynomial. For example, in x³ + y³ + 3xy(x + y), the degree of the polynomial is 3. For a non-zero constant polynomial, the degree is zero. Apart from these, there are other types of polynomials such as:

• Linear polynomial – of degree one
• Quadratic Polynomial- of degree two
• Cubic Polynomial – of degree three

This topic has been widely discussed in class 9 and class 10.

Polynomials in One Variable

Polynomials in one variable are expressions which consist of only one type of variable in the entire expression.
Example of polynomials in one variable:

• 3a
• 2x² + 5x + 15

Zeroes of Polynomial

The zeroes of polynomials are the points where the polynomial is equal to 0 as a whole.

Remainder Theorem

If p(x) is any polynomial having degree greater than or equal to 1, and if it is divided by the linear polynomial x – a, then the remainder is p(a).

Factor Theorem

x – c is a factor of the polynomial p(x), if p(c) = 0. Also, if x – c is a factor of p(x), then p(c) = 0.

Factorisation of Polynomials

Factorisation of polynomials is the process of expressing the polynomials as the product of two or more polynomials.
For example, the polynomial x²-x-6 can be factorised as (x-3)(x+2)

Also read: Factorisation of Polynomials

Algebraic Identities

Algebraic identities are algebraic equations which are valid for all values. The important algebraic identities used in Class 9 Maths chapter 2 polynomials are listed below:(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)

Polynomials Class 9 Examples

Example 1:

Write the coefficients of x in each of the following:

• 3x + 1
• 23x² – 5x + 1

Solution:

In 3x + 1, the coefficient of x is 3.

In 23x² – 5x + 1, the coefficient of x is -5.

Example 2:

What are the degrees of following polynomials?

1. 3a² + a – 1
2. 32x³ + x – 1

Solution: 

1. 3a² + a – 1 : The degree is 2
2. 32x³ + x – 1 : The degree is 3

Polynomials Class 9 Important Questions

1. Find value of polynomial 2x² + 5x + 1 at x = 3.
2. Check whether x = -1/6 is zero of the polynomial p(a) = 6a + 1.
3. Divide 3a² + x – 1 by a + 1.
4. Find value of k, if (a – 1) is the factor of p(a) = ka² – 3a + k.
5. Factorise each of the following:
   • 4x² + 9y² + 16z² + 12xy – 24yx – 16xz
   • 2x² + y² + 8z² – 2√2xy + 4√2yz – 8xz