Chapter 2 - Polynomials Revision Notes

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 in it 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 binomial. Example: 5x + 2.

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

Constant Polynomial

The real numbers can also be expressed as polynomials. Like 3, 6, 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 as 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

Polynomials in One Variable

Polynomials in one variable are the 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 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 polynomails.

For example, the polynomial x²-x-6 can be factorised as (x-3)(x+2)

Algebraic Identities

The algebraic identities are the algebraic equations, which is 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 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