[PDF] Class 9th Chapter 2 Polynomial Revision Notes & Solution

101386_6chapter_2_polynomial.pdf Class 9th Chapter 2 Polynomial Revision Notes & Solution www.free-education.in Page 1

Polynomials Class 9 Topics

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. 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. Each term of the polynomial has a coefficient. For

example, if 2x + 1 is the polynomial, then the coefficient of x is 2.

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.

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.

Example of polynomials are:

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

Quadratic Equation Algebraic Identities

Quadratic Formula & Quadratic Polynomial Degree Of A Polynomial

Polynomials in One Variable

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

Class 9th Chapter 2 Polynomial Revision Notes & Solution www.free-education.in Page 2

Example of polynomials in one variable:

3a 2x2 + 5x + 15

Polynomial 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 note here provides a brief of the chapter so that students

find it easy to have a glance at once. The key points covered in the chapter have been noted. Go through the

points and solve problems based on them. Some important points in Polynomials Class 9 Chapter 2 are given below:

An algebraic expression p(x) = a0xn + a1xn-1 + a2xn-2 н ͙ an is a polynomial where a0, a1, ͙͙͙. an are real numbers

and n is non-negative integer. A term is either a variable or a single number or it can be a combination of variable and numbers. The degree of the polynomial is the highest power of the variable in a polynomial. A polynomial of degree 1 is called as a linear polynomial. A polynomial of degree 2 is called a quadratic polynomial. A polynomial of degree 3 is called a cubic polynomial. A polynomial of 1 term is called a monomial. A polynomial of 2 terms is called binomial. A polynomial of 3 terms is called a trinomial.

A real number ͚a͛ is a zero of a polynomial p(dž) if p(a) с 0, where a is also known as root of the equation p(x) =

0.

A linear polynomial in one variable has a unique zero, a polynomial of a non-zero constant has no zero, and

each real number is a zero of the zero polynomial.

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.

The degree of the zero polynomial is not defined. (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

Class 9th Chapter 2 Polynomial Revision Notes & Solution www.free-education.in Page 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 - 2я2džy н 4я2yz t 8xz Access Answers of Maths NCERT class 9 Chapter 2 - Polynomials

Exercise 2.1 Page: 32

1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your

answer. (i) 4x2-3x+7

Solution:

The equation 4x2-3x+7 can be written as 4x2-3x1+7x0

Since x is the only variable in the given equation and the powers of x (i.e., 2, 1 and 0) are whole numbers, we can

say that the expression 4x2-3x+7 is a polynomial in one variable. (ii) y2ня2

Solution:

The equation y2+я2 can be written as y2+я2y0

Since y is the only variable in the given equation and the powers of y (i.e., 2 and 0) are whole numbers, we can say

that the expression y2+я2 is a polynomial in one variable. (iii) 3яtнtя2

Solution:

The eƋuation 3яtн