# NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials - A polynomial is an algebraic expression which consists of variables and coefficient with operations such as additions, subtraction, multiplication, and non-negative exponents. In this particular chapter, you will learn the operations of two or more polynomials. NCERT solutions for class 9 maths chapter 2 Polynomials are there to help you while solving the problems related to this particular chapter. NCERT class 9 Polynomials, introduces a lot of important concepts that will be helpful for those students who are targeting exams like JEE, CAT, SSC, etc. It is an important topic in maths that comes under the algebra unit which holds 20 marks in the CBSE class 9 maths final exams. In this particular chapter, you will study the definition of a polynomial, zeroes, coefficient, degrees, and terms of a polynomial, type of a polynomial. You will also study the remainder and factor theorems and the factorization of polynomials. In Polynomials, there are a total of 5 exercises that comprise of a total of 33 questions. Solutions of NCERT class 9 maths chapter 2 Polynomials will cover the detailed solution to each and every question present in the practice exercises including optional exercises. CBSE NCERT solutions for class 9 maths chapter 2 Polynomials can also be used while doing homework. It can be a good tool for the class 9 students as it is designed in such a manner so that a student can fetch the maximum marks available for the particular question. NCERT solutions are also available for other classes and subjects which can be downloaded by clicking on the link given.

## NCERT solutions for class 9 maths chapter 2 Polynomials Excercise: 2.1

### Q1 (i) Is the following expression polynomial in one variable? State reasons for your answer.

YES
Given polynomial  has only one variable which is x

YES
Given polynomial has only one variable which is y

NO
Because we can observe that the exponent of variable t in term  is    which is not a whole number.
Therefore this expression is not a polynomial.

NO
Because we can observe that the exponent of variable y in term  is  which is not a whole number. Therefore this expression is not a polynomial.

NO
Because in the given polynomial   there are 3 variables which are x, y, t. That's why this is polynomial in three variable not in one variable.

Coefficient of  in polynomial    is  1.

Coefficient of  in  polynomial    is  -1.

Coefficient of    in polynomial    is

Coefficient of    in polynomial     is  0

Degree of a polynomial is the highest power of the variable in the polynomial.
In binomial, there are two terms
Therefore,  binomial of degree 35 is
Eg:-
In monomial, there is only one term in it.
Therefore, monomial of degree 100 can be written as

Degree of a polynomial is the highest power of the variable in the polynomial.
Therefore, the degree of polynomial    is  3.

Degree of a polynomial is the highest power of the variable in the polynomial.

Therefore, the degree of polynomial    is  2.

Degree of a polynomial is the highest power of the variable in the polynomial.

Therefore, the degree of polynomial    is  1

Degree of a polynomial is the highest power of the variable in the polynomial.

In this case, only a constant value 3 is there and the degree of a constant polynomial is always 0.

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 2

Therefore, it is a quadratic polynomial.

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 3

Therefore, it is a cubic polynomial

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 2

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 1

Therefore, it is linear polynomial

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 1

Therefore, it is linear polynomial

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 2

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

Given polynomial is    with degree 3

Therefore, it is a cubic polynomial

## NCERT solutions for class 9 maths chapter 2 Polynomials Excercise: 2.2

### Q1 (i) Find the value of the polynomial at

Given polynomial is

Now, at   value is

Therefore, value of polynomial  at x = 0 is  3

Given polynomial is

Now, at   value is

Therefore, value of polynomial  at x = -1 is  -6

Given polynomial is

Now, at   value is

Therefore, value of polynomial  at x = 2 is  -3

Given polynomial is

Now,

Therefore, valures of p(0), p(1) and p(2) are 1 , 1 and 3 respectively.

Given polynomial is

Now,

Therefore, values of p(0), p(1) and p(2) are 2 , 4 and 4 respectively

Given polynomial is

Now,

Therefore, values of p(0), p(1) and p(2) are 0 , 1 and 8 respectively

Given polynomial is

Now,

Therefore, values of p(0), p(1) and p(2) are -1 , 0 and 3 respectively

Given polynomial is

Now, at     it's value is

Therefore, yes   is a zero of  polynomial

Given polynomial is

Now, at     it's value is

Therefore, no     is not a zero of  polynomial

Given polynomial is

Now, at  x = 1   it's value is

And at x = -1

Therefore, yes  x = 1 , -1   are zeros of  polynomial

Given polynomial is

Now, at  x = 2   it's value is

And at x = -1

Therefore, yes  x = 2 , -1   are zeros of  polynomial

Given polynomial is

Now, at  x = 0   it's value is

Therefore, yes  x = 0  is a zeros of  polynomial

Given polynomial is

Now, at     it's value is

Therefore, yes     is a zeros of  polynomial

Given polynomial is

Now, at     it's value is

And at

Therefore,     is a zeros of polynomial

whereas   is not a zeros of  polynomial

Given polynomial is

Now, at     it's value is

Therefore,     is not a zeros of  polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore, x = -5 is the zero of polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore, x = 5 is a zero of polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore,  is a zero of polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore,   is a zero of polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore,   is a zero of polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore,   is a zero of polynomial

Given polynomial is

Zero of a polynomial is that value of the variable at which the value of the polynomial is obtained as 0.

Now,

Therefore,    is a zero of polynomial

## NCERT solutions for class 9 maths chapter 2 Polynomials Excercise: 2.3

### Q1 (i) Find the remainder when  is divided by

When we divide    by

By long division method, we will get Therefore, remainder is  .

## Q1 (ii) Find the remainder when  is divided by

When we divide    by

By long division method, we will get Therefore, the remainder is

When we divide    by

By long division method, we will get Therefore, remainder is  .

When we divide    by

By long division method, we will get Therefore, the remainder is

When we divide    by

By long division method, we will get Therefore, the remainder is

When we divide    by

By long division method, we will get Therefore, remainder is

When we divide    by

We can also write    as

By long division method, we will get Since, remainder is not equal to  0

Therefore,    is not a factore of

## NCERT solutions for class 9 maths chapter 2 Polynomials Excercise: 2.4

Zero of polynomial     is -1.

If    is a factor of polynomial

Then,   must be equal to zero

Now,

Therefore,    is a factor of polynomial

Zero of polynomial     is -1.

If    is a factor of polynomial

Then,   must be equal to zero

Now,

Therefore,    is not a factor of polynomial

Zero of polynomial     is -1.

If    is a factor of polynomial

Then,   must be equal to zero

Now,

Therefore,    is not a factor of polynomial

Zero of polynomial     is -1.

If    is a factor of polynomial

Then,   must be equal to zero

Now,

Therefore,    is not a factor of polynomial

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,     is factor of polynomial

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,     is not a factor of polynomial

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,     is a factor of polynomial

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,  value of k is

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,  value of k is

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,  value of k is

Zero of polynomial    is

If    is factor of polynomial

Then,    must be equal to zero

Now,

Therefore,  value of k is

Q4 (i) Factorise :

Given polynomial is

We need to factorise the middle term into two terms such that their product is equal to  and their sum is equal to

We can solve it as

Q4 (ii) Factorise :

Given polynomial is

We need to factorise the middle term into two terms such that their product is equal to  and their sum is equal to

We can solve it as

Q4 (iii) Factorise :

Given polynomial is

We need to factorise the middle term into two terms such that their product is equal to  and their sum is equal to

We can solve it as

Q4 (iv) Factorise :

Given polynomial is

We need to factorise the middle term into two terms such that their product is equal to  and their sum is equal to

We can solve it as

Q5 (i) Factorise :

Given polynomial is

Now, by hit and trial method we observed that    is one of the factors of the given polynomial.

By long division method, we will get We know that  Dividend = (Divisor  Quotient) + Remainder

Therefore, on factorization of   we will get

Q5 (ii) Factorise :

Given polynomial is

Now, by hit and trial method we observed that    is one of the factore of given polynomial.

By long division method, we will get We know that  Dividend = (Divisor  Quotient) + Remainder

Therefore, on factorization of   we will get

Q5 (iii) Factorise :

Given polynomial is

Now, by hit and trial method we observed that    is one of the factore of given polynomial.

By long division method, we will get We know that  Dividend = (Divisor  Quotient) + Remainder

Therefore, on factorization of   we will get

Q5 (iv) Factorise :

Given polynomial is

Now, by hit and trial method we observed that    is one of the factors of the given polynomial.

By long division method, we will get We know that  Dividend = (Divisor  Quotient) + Remainder

Therefore, on factorization of   we will get

## NCERT solutions for class 9 maths chapter 2 Polynomials Excercise: 2.5

We will use identity

Put

Therefore,   is equal to

We will use identity

Put

Therefore,   is equal to

We can write    as

We will use identity

Put

Therefore,   is equal to

We will use identity

Put

Therefore,   is equal to

We can write   as

We will use identity

Put

Therefore,   is equal to

We can rewrite    as

We will use identity

Put

Therefore, value of   is

We can rewrite    as

We will use identity

Put

Therefore, value of    is

We can rewrite    as

We will use identity

Put

Therefore, value of    is

We can rewrite  as

Using identity

Here,

Therefore,

We can rewrite  as

Using identity

Here,

Therefore,

We can rewrite     as

Using identity

Here,

Therefore,

Given is

We will Use identity

Here,

Therefore,

Given is

We will Use identity

Here,

Therefore,

Given is

We will Use identity

Here,

Therefore,

Given is

We will Use identity

Here,

Therefore,

Given is

We will Use identity

Here,

Therefore,

Given is

We will Use identity

Here,

Therefore,

Q5 (i) Factorise:

We can rewrite    as

We will Use identity

Here,

Therefore,

Q5 (ii) Factorise:

We can rewrite    as

We will Use identity

Here,

Therefore,

Given is

We will use identity

Here,

Therefore,

Given is

We will use identity

Here,

Therefore,

Given is

We will use identity

Here,

Therefore,

Given is

We will use identity

Here,

Therefore,

We can rewrite     as

We will use identity

Here,

Therefore,

We can rewrite     as

We will use identity

Here,

Therefore,

We can rewrite     as

We will use identity

Here,

Therefore,

Q8 (i) Factorise the following:

We can rewrite     as

We will use identity

Here,

Therefore,

Q8 (ii) Factorise the following:

We can rewrite     as

We will use identity

Here,

Therefore,

Q8 (iii) Factorise the following:

We can rewrite     as

We will use identity

Here,

Therefore,

Q8 (iv) Factorise the following:

We can rewrite     as

We will use identity

Here,

Therefore,

Q8 (v) Factorise the following:

We can rewrite     as

We will use identity

Here,

Therefore,

Q9 (i) Verify:

We know that

Now,

Hence proved.

Q9 (ii) Verify:

We know that

Now,

Hence proved.

Q10 (i) Factorise the following:

We know that

Now, we can write    as

Here,

Therefore,

Q10 (ii) Factorise the following:

We know that

Now, we can write    as

Here,

Therefore,

Q11 Factorise:

Given is

Now, we know that

Now, we can write    as

Here,

Therefore,

Q12 Verify that

We know that

Now, multiply and divide the R.H.S. by 2

Hence proved.

We know that

Now, It is given that

Therefore,

Hence proved.

Given is

We know that

If      then ,

Here,

Therefore,

Therefore, value of    is

Given is

We know that

If      then ,

Here,

Therefore,

Therefore, value of    is

We know that

Area of rectangle is =

It is given that area  =

Now, by splitting middle term method

case (i) :-   Length =   and  Breadth =

case (ii) :-   Length =   and  Breadth =

We know that

Area of rectangle is =

It is given that area  =

Now, by splitting the middle term method

case (i) :-   Length =   and  Breadth =

case (ii) :-   Length =   and  Breadth =

 Volume :

We know that

Volume of cuboid is =

It is given that volume  =

Now,

Therefore,one of the possible answer is possible

Length =   and  Breadth =   and  Height =

 Volume :

We know that

Volume of cuboid is =

It is given that volume  =

Now,

Therefore,one of the possible answer is possible

Length =   and  Breadth =   and  Height =

## NCERT solutions for class 9 maths chapter wise

 Chapter No. Chapter Name Chapter 1 NCERT solutions for class 9 maths chapter 1 Number Systems Chapter 2 CBSE NCERT solutions for class 9 maths chapter 2 Polynomials Chapter 3 Solutions of NCERT class 9 maths chapter 3 Coordinate Geometry Chapter 4 NCERT solutions for class 9 maths chapter 4 Linear Equations In Two Variables Chapter 5 CBSE NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry Chapter 6 Solutions of NCERT class 9 maths chapter 6 Lines And Angles Chapter 7 NCERT solutions for class 9 maths chapter 7 Triangles Chapter 8 CBSE NCERT solutions for class 9 maths chapter 8 Quadrilaterals Chapter 9 Solutions of NCERT class 9 maths chapter 9 Areas of Parallelograms and Triangles Chapter 10 NCERT solutions for class 9 maths chapter 10 Circles Chapter 11 CBSE NCERT solutions for class 9 maths chapter 11 Constructions Chapter 12 Solutions of NCERT class 9 maths chapter 12 Heron’s Formula Chapter 13 NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Chapter 14 CBSE NCERT solutions for class 9 maths chapter 14 Statistics Chapter 15 Solutions of NCERT class 9 maths chapter 15 Probability

## How to use NCERT solutions for class 9 maths chapter 2 Polynomials?

• First of all, learn some basics and concepts regarding chapter polynomials.
• While reading the basics, go through the examples so that you can understand the applications of the concepts.
• Once you have done the above two points, then you can directly move to the practice exercises.
• While practicing the exercises, if you stuck anywhere then you can take the help of the NCERT solutions for class 9 maths chapter 2 Polynomials.
• After the completion of practice exercises, you can go through some previous year question papers.

Keep Working Hard and Happy Learning!