# 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. NCERT solutions for class 9 maths chapter 2 Polynomials will cover the detailed solution to each and every question present in the practice exercises including optional exercises. 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. Here you will get NCERT solutions for class 9 also.

## ** 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.

** Answer: **

** YES **

Given polynomial has only one variable which is ** x **

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

** Answer: **

** YES **

Given polynomial has only one variable which is ** y **

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

** Answer: **

** 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.

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

** Answer: **

** 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.

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

** Answer: **

** 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.

** Q3 ** Give one example each of a binomial of degree 35, and of a monomial of degree 100.

** Answer: **

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

** Q4 (i) ** Write the degree the following polynomial:

** Answer: **

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

Therefore, the degree of polynomial is ** 3 ** .

** Q4 (ii) ** Write the degree the following polynomial:

** Answer: **

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

Therefore, the degree of polynomial is ** 2. **

** Q4 (iii) ** Write the degree the following polynomial:

** Answer: **

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

Therefore, the degree of polynomial is ** 1 **

** Q4 (iv) ** Write the degree the following polynomial: 3

** Answer: **

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. **

** Q5 (i) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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.

** Q5 (ii) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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

** Q5 (iii) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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 quadratic polynomial.

** Q5 (iv) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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

** Q5 (v) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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

** Q5 (vi) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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 quadratic polynomial

** Q5 (vii) ** Classify the following as linear, quadratic and cubic polynomial:

** Answer: **

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

** Answer: **

Given polynomial is

Now, at value is

Therefore, value of polynomial at x = 0 is ** 3 **

** Q1 (ii) ** Find the value of the polynomial at

** Answer: **

Given polynomial is

Now, at value is

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

** Q1 (iii) ** Find the value of the polynomial at

** Answer: **

Given polynomial is

Now, at value is

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

** Q2 (i) ** Find * p(0) * , * p(1) * and * p(2) * for each of the following polynomials:

** Answer: **

Given polynomial is

Now,

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

** Q2 (ii) ** Find * p(0) * , * p(1) * and * p(2) * for each of the following polynomials:

** Answer: **

Given polynomial is

Now,

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

** Q2 (iii) ** Find p(0), p(1) and p(2) for each of the following polynomials:

** Answer: **

Given polynomial is

Now,

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

** Q2 (iv) ** Find p(0), p(1) and p(2) for each of the following polynomials:

** Answer: **

Given polynomial is

Now,

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

** Q3 (i) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at it's value is

Therefore, yes is a zero of polynomial

** Q3 (ii) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at it's value is

Therefore, no is not a zero of polynomial

** Q3 (iii) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at ** x = 1 ** it's value is

And at ** x = -1 **

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

** Q3 (iv) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at ** x = 2 ** it's value is

And at ** x = -1 **

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

** Q3 (v) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at ** x = 0 ** it's value is

Therefore, yes ** x = 0 ** is a zeros of polynomial

** Q3 (vi) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at it's value is

Therefore, yes is a zeros of polynomial

** Q3 (vii) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at it's value is

And at

Therefore, is a zeros of polynomial .

whereas is not a zeros of polynomial

** Q3 (viii) ** Verify whether the following are zeroes of the polynomial, indicated against it.

** Answer: **

Given polynomial is

Now, at it's value is

Therefore, is not a zeros of polynomial

** Q4 (i) ** Find the zero of the polynomial in each of the following cases:

** Answer: **

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

** Q4 (ii) ** Find the zero of the polynomial in each of the following cases:

** Answer: **

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

** Q4 (iii) ** Find the zero of the polynomial in each of the following cases:

** Answer: **

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

** Q4 (iv) ** Find the zero of the polynomial in each of the following cases:

** Answer: **

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

** Q4 (v) ** Find the zero of the polynomial in each of the following cases:

** Answer: **

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

** Q4 (vi) ** Find the zero of the polynomial in each of the following cases:

** Answer: **

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

** Q4 (vii) ** Find the zero of the polynomial in each of the following cases: are real numbers

** Answer: **

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

** Answer: **

When we divide by .

By long division method, we will get

Therefore, remainder is .

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

** Answer: **

When we divide by .

By long division method, we will get

Therefore, the remainder is

** Q1 (iii) ** Find the remainder when is divided by

** Answer: **

When we divide by .

By long division method, we will get

Therefore, remainder is .

** Q1 (iv) ** Find the remainder when is divided by

** Answer: **

When we divide by .

By long division method, we will get

Therefore, the remainder is

** Q1 (v) ** Find the remainder when is divided by

** Answer: **

When we divide by .

By long division method, we will get

Therefore, the remainder is

** Q2 ** Find the remainder when is divided by .

** Answer: **

When we divide by .

By long division method, we will get

Therefore, remainder is

** Q3 ** Check whether is a factor of

** Answer: **

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 factor of

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

** Q1 (i) ** Determine which of the following polynomials has a factor :

** Answer: **

Zero of polynomial is ** -1. **

If is a factor of polynomial

Then, must be equal to zero

Now,

Therefore, is a factor of polynomial

** Q1 (ii) ** Determine which of the following polynomials has a factor :

** Answer: **

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

** Q1 (iii) ** Determine which of the following polynomials has a factor :

** Answer: **

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

** Q1 (iv) ** Determine which of the following polynomials has a factor :

** Answer: **

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

** Q2 (i) ** Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following case:

** Answer: **

Zero of polynomial is

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, is factor of polynomial

** Q2 (ii) ** Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following case:

** Answer: **

Zero of polynomial is

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, is not a factor of polynomial

** Q2 (iii) ** Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following case:

** Answer: **

Zero of polynomial is

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, is a factor of polynomial

** Q3 (i) ** Find the value of * k * , if is a factor of p(x) in the following case:

** Answer: **

Zero of polynomial is

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, value of k is

** Q3 (ii) ** Find the value of * k * , if is a factor of p(x) in the following case:

** Answer: **

Zero of polynomial is

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, value of k is

** Q3 (iii) ** Find the value of * k * , if is a factor of p(x) in the following case:

** Answer: **

Zero of polynomial is

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, value of k is

** Q3 (iv) ** the value of * k * , if is a factor of p(x) in the following case:

** Answer: **

If is factor of polynomial

Then, must be equal to zero

Now,

Therefore, value of k is

** Q4 (i) ** Factorise :

** Answer: **

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 :

** Answer: **

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 :

** Answer: **

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 :

** Answer: **

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 :

** Answer: **

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 :

** Answer: **

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 (iii) ** Factorise :

** Answer: **

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 :

** Answer: **

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 **

** Q1 (i) ** Use suitable identities to find the following product:

** Answer: **

We will use identity

Put

Therefore, is equal to

** Q1 (ii) ** Use suitable identities to find the following product:

** Answer: **

We will use identity

Put

Therefore, is equal to

** Q1 (iii) ** Use suitable identities to find the following product:

** Answer: **

We can write as

We will use identity

Put

Therefore, is equal to

** Q1 (iv) ** Use suitable identities to find the following product:

** Answer: **

We will use identity

Put

Therefore, is equal to

** Q1 (v) ** Use suitable identities to find the following product:

** Answer: **

We can write as

We will use identity

Put

Therefore, is equal to

** Q2 (i) ** Evaluate the following product without multiplying directly:

** Answer: **

We can rewrite as

We will use identity

Put

Therefore, value of is

** Q2 (ii) ** Evaluate the following product without multiplying directly:

** Answer: **

We can rewrite as

We will use identity

Put

Therefore, value of is

** Q2 (iii) ** Evaluate the following product without multiplying directly:

** Answer: **

We can rewrite as

We will use identity

Put

Therefore, value of is

** Q3 (i) ** Factorise the following using appropriate identities:

** Answer: **

We can rewrite as

Using identity

Here,

Therefore,

** Q3 (ii) ** Factorise the following using appropriate identities:

** Answer: **

We can rewrite as

Using identity

Here,

Therefore,

** Q3 (iii) ** Factorise the following using appropriate identities:

** Answer: **

We can rewrite as

Using identity

Here,

Therefore,

** Q4 (i) ** Expand each of the following, using suitable identities:

** Answer: **

Given is

We will Use identity

Here,

Therefore,

** Q4 (ii) ** Expand each of the following, using suitable identities:

** Answer: **

Given is

We will Use identity

Here,

Therefore,

** Q4 (iii) ** Expand each of the following, using suitable identities:

** Answer: **

Given is

We will Use identity

Here,

Therefore,

** Q4 (iv) ** Expand each of the following, using suitable identities:

** Answer: **

Given is

We will Use identity

Here,

Therefore,

** Q4 (v) ** Expand each of the following, using suitable identities:

** Answer: **

Given is

We will Use identity

Here,

Therefore,

** Q4 (vi) ** Expand each of the following, using suitable identities:

** Answer: **

Given is

We will Use identity

Here,

Therefore,

** Q6 (i) ** Write the following cubes in expanded form:

** Answer: **

Given is

We will use identity

Here,

Therefore,

** Q6 (ii) ** Write the following cube in expanded form:

** Answer: **

Given is

We will use identity

Here,

Therefore,

** Q6 (iii) ** Write the following cube in expanded form:

** Answer: **

Given is

We will use identity

Here,

Therefore,

** Q6 (iv) ** Write the following cube in expanded form:

** Answer: **

Given is

We will use identity

Here,

Therefore,

** Q7 (i) ** Evaluate the following using suitable identities:

** Answer: **

We can rewrite as

We will use identity

Here,

Therefore,

** Q7 (ii) ** Evaluate the following using suitable identities:

** Answer: **

We can rewrite as

We will use identity

Here,

Therefore,

** Q7 (iii) ** Evaluate the following using suitable identities:

** Answer: **

We can rewrite as

We will use identity

Here,

Therefore,

** Q11 ** Factorise:

** Answer: **

Given is

Now, we know that

Now, we can write as

Here,

Therefore,

** Q14 (i) ** Without actually calculating the cubes, find the value of each of the following:

** Answer: **

Given is

We know that

If then ,

Here,

Therefore,

Therefore, value of is

** Q14 (ii) ** Without actually calculating the cubes, find the value of the following:

** Answer: **

Given is

We know that

If then ,

Here,

Therefore,

Therefore, value of is

** Answer: **

We know that

Area of rectangle is =

It is given that area =

Now, by splitting middle term method

Therefore, two answers are possible

** case (i) :- ** Length = and Breadth =

** case (ii) :- ** ** ** Length = and Breadth =

** Answer: **

We know that

Area of rectangle is =

It is given that area =

Now, by splitting the middle term method

Therefore, two answers are possible

** case (i) :- ** Length = and Breadth =

** case (ii) :- ** ** ** Length = and Breadth =

** Q16 (i) ** What are the possible expressions for the dimensions of the cuboid whose volumes is given below?

Volume : |

** Answer: **

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 =

** Q16 (ii) ** What are the possible expressions for the dimensions of the cuboid whose volumes is given below?

Volume : |

** Answer: **

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 **

** NCERT solutions for class 9 subject wise **

** 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 practising 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! *

## Frequently Asked Question (FAQs) - NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

**Question: **What are the important topics in chapter Polynomials ?

**Answer: **

Definition of a polynomial, zeroes, coefficient, degrees, and terms of a polynomial, different types of a polynomial, remainder and factor theorems, and the factorization of polynomials are the important topics of this chapter.

**Question: **Does NCERT class 9 maths is tough ?

**Answer: **

NCERT maths for class 9 is not tough at all, some students find it tough because in the previous classes the math we study is very simple.

**Question: **How many chapters are there in the CBSE class 9 maths ?

**Answer: **

There are 15 chapters starting from the number system to probability in the CBSE class 9 maths.

**Question: **Does CBSE provides the solutions of NCERT for class 9 maths ?

**Answer: **

No, CBSE doesn’t provide NCERT solutions for any class or subject.

**Question: **Where can I find the complete solutions of NCERT for class 9 maths ?

**Answer: **

Here you will get the detailed NCERT solutions for class 9 maths by clicking on the link.

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