NCERT solutions for Mathematics Class 9 chapter 2 - Polynomials [Latest edition]

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Solutions for Chapter 2: Polynomials

Below listed, you can find solutions for Chapter 2 of CBSE NCERT for Mathematics Class 9.


Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5
Exercise 2.1 [Page 32]

NCERT solutions for Mathematics Class 9 Chapter 2 Polynomials Exercise 2.1 [Page 32]

Exercise 2.1 | Q 1. (i) | Page 32

Which of the following expression is polynomial in one variable and which is not? State the reason for your answer.

4x2 - 3x + 7

Exercise 2.1 | Q 1. (ii) | Page 32

Which of the following expression is polynomial in one variable and which is not? State the reason for your answer.

`y^2+sqrt2`

Exercise 2.1 | Q 1. (iii) | Page 32

Which of the following expression is polynomial in one variable and which is not? State the reason for your answer.

`3sqrtt+tsqrt2`

Exercise 2.1 | Q 1. (iv) | Page 32

Which of the following expression is polynomial in one variable and which is not? State the reason for your answer.

`y+2/y`

Exercise 2.1 | Q 1. (v) | Page 32

Which of the following expression is polynomial in one variable and which is not? State the reason for your answer.

`x^10+y^3+t^50`

Exercise 2.1 | Q 2.1 | Page 32

Write the coefficient of x2 in the following:-

`2+x^2+x`

Exercise 2.1 | Q 2.2 | Page 32

Write the coefficient of x2 in the following:-

`2-x^2+x^3`

Exercise 2.1 | Q 2.3 | Page 32

Write the coefficient of x2 in the following:-

`pi/2x^2+x`

Exercise 2.1 | Q 2.4 | Page 32

Write the coefficient of x2 in the following:-

`sqrt2x-1`

Exercise 2.1 | Q 3 | Page 32

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

Exercise 2.1 | Q 4.1 | Page 32

Write the degree of the following polynomial:-

5x3 + 4x2 +7x

Exercise 2.1 | Q 4.2 | Page 32

Write the degree of the following polynomial:-

4 - y2

Exercise 2.1 | Q 4.3 | Page 32

Write the degree of the following polynomials:-

`5t - sqrt7`

Exercise 2.1 | Q 4.4 | Page 32

Write the degree of each of the following polynomials:-

3

Exercise 2.1 | Q 5. (i) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

x2 + x

Exercise 2.1 | Q 5. (ii) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

 x - x3

Exercise 2.1 | Q 5. (iii) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

y + y2 + 4

Exercise 2.1 | Q 5. (iv) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

1 + x

Exercise 2.1 | Q 5. (v) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

3t

Exercise 2.1 | Q 5. (vi) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

r2

Exercise 2.1 | Q 5. (vii) | Page 32

Classify the following as linear, quadratic, and cubic polynomials:

7x3

Exercise 2.2 [Pages 34 - 35]

NCERT solutions for Mathematics Class 9 Chapter 2 Polynomials Exercise 2.2 [Pages 34 - 35]

Exercise 2.2 | Q 1.1 | Page 34

Find the value of the polynomial 5x – 4x2 + 3 at

x = 0

Exercise 2.2 | Q 1.2 | Page 34

Find the value of the polynomial 5x – 4x2 + 3 at

x = –1

Exercise 2.2 | Q 1.3 | Page 34

Find the value of the polynomial 5x – 4x2 + 3 at

 x = 2

Exercise 2.2 | Q 2.1 | Page 34

Find p(0), p(1) and p(2) for the following polynomials:-

p(y) = y2 – y + 1

Exercise 2.2 | Q 2.2 | Page 34

Find p(0), p(1) and p(2) for the following polynomials:-

p(t) = 2 + t + 2t2 – t3

Exercise 2.2 | Q 2.3 | Page 34

Find p(0), p(1) and p(2) for the following polynomials:-

p(x) = x3

Exercise 2.2 | Q 2.4 | Page 34

Find p(0), p(1) and p(2) for the following polynomials:-

p(x) = (x – 1) (x + 1)

Exercise 2.2 | Q 3. (i) | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

`p(x) = 3x + 1; x = -1/3`

Exercise 2.2 | Q 3. (ii) | Page 35

Verify whether the following zeroes of the polynomial are indicated against them.

p(x) = 5x – π, `x = 4/5`

Exercise 2.2 | Q 3.3 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = x2 – 1, x = 1, –1

Exercise 2.2 | Q 3.4 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = (x + 1) (x – 2), x = – 1, 2

Exercise 2.2 | Q 3.5 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = x2, x = 0

Exercise 2.2 | Q 3.6 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = lx + m, `x = – m/l`

Exercise 2.2 | Q 3.7 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

`p(x) = 3x^2 – 1, x = -1/sqrt3,2/sqrt3`

Exercise 2.2 | Q 3.8 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = 2x + 1, `x = 1/2`

Exercise 2.2 | Q 4.1 | Page 35

Find the zero of the polynomial in the following case:- p(x) = x + 5

Exercise 2.2 | Q 4.2 | Page 35

Find the zero of the polynomial in the following case:- p(x) = x – 5

Exercise 2.2 | Q 4.3 | Page 35

Find the zero of the polynomial in the following case:- p(x) = 2x + 5

Exercise 2.2 | Q 4.4 | Page 35

Find the zero of the polynomial in the following case:- p(x) = 3x – 2

Exercise 2.2 | Q 4.5 | Page 35

Find the zero of the polynomial in the following case:- p(x) = 3x

Exercise 2.2 | Q 4.6 | Page 35

Find the zero of the polynomial in the following case:-

p(x) = ax, a ≠ 0

Exercise 2.2 | Q 4.7 | Page 35

Find the zero of the polynomial in the following case:- p(x) = cx + d, c ≠ 0, c, d are real numbers.

Exercise 2.3 [Page 40]

NCERT solutions for Mathematics Class 9 Chapter 2 Polynomials Exercise 2.3 [Page 40]

Exercise 2.3 | Q 1.1 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x+1.

Exercise 2.3 | Q 1.2 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by `x - 1/2`

Exercise 2.3 | Q 1.3 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x.

Exercise 2.3 | Q 1.4 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x + π.

Exercise 2.3 | Q 1.5 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by 5 + 2x.

Exercise 2.3 | Q 2 | Page 40

Find the remainder when x3 – ax2 + 6x – a is divided by x – a.

Exercise 2.3 | Q 3 | Page 40

Check whether 7 + 3x is a factor of 3x3 + 7x.

Exercise 2.4 [Pages 43 - 44]

NCERT solutions for Mathematics Class 9 Chapter 2 Polynomials Exercise 2.4 [Pages 43 - 44]

Exercise 2.4 | Q 1.1 | Page 43

Determine if the following polynomial has (x + 1) a factor :-

x3 + x2 + x + 1

Exercise 2.4 | Q 1.2 | Page 43

Determine if the following polynomial has (x + 1) a factor :-

x4 + x3 + x2 + x + 1

Exercise 2.4 | Q 1.3 | Page 43

Determine which of the following polynomials has (x + 1) a factor :-

x4 + 3x3 + 3x2 + x + 1

Exercise 2.4 | Q 1.4 | Page 43

Determine which of the following polynomials has (x + 1) a factor :-
`x^3-x^2-(2+sqrt2)x+sqrt2`

Exercise 2.4 | Q 2.1 | Page 44

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

p(x) = 2x3 + x2 – 2x – 1, g(x) = x + 1

Exercise 2.4 | Q 2.2 | Page 44

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

p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2

Exercise 2.4 | Q 2.3 | Page 44

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

p(x) = x3 − 4x2 + x + 6, g(x) = x − 3

Exercise 2.4 | Q 3.1 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:- p(x) = x2 + x + k

Exercise 2.4 | Q 3.2 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:-

`p(x) = 2x^2+kx+sqrt2`

 

Exercise 2.4 | Q 3.3 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:- `p(x) = kx^2 - sqrt2x +1`

 

Exercise 2.4 | Q 3.4 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:-

p(x) = kx2 – 3x + k

Exercise 2.4 | Q 4.1 | Page 44

Factorise :- 12x2 – 7x + 1

Exercise 2.4 | Q 4.2 | Page 44

Factorise :- 2x2 + 7x + 3

Exercise 2.4 | Q 4.3 | Page 44

Factorise :- 6x2 + 5x – 6

Exercise 2.4 | Q 4.4 | Page 44

Factorise :- 3x2 – x – 4

Exercise 2.4 | Q 5.1 | Page 44

Factorise :- x3 – 2x2 – x + 2

Exercise 2.4 | Q 5.2 | Page 44

Factorise :- x3 – 3x2 – 9x – 5

Exercise 2.4 | Q 5.3 | Page 44

Factorise :- x3 + 13x2 + 32x + 20

Exercise 2.4 | Q 5.4 | Page 44

Factorise :- 2y3 + y2 – 2y – 1

Exercise 2.5 [Pages 48 - 50]

NCERT solutions for Mathematics Class 9 Chapter 2 Polynomials Exercise 2.5 [Pages 48 - 50]

Exercise 2.5 | Q 1.1 | Page 48

Use suitable identities to find the following products :- (x + 4) (x + 10) 

Exercise 2.5 | Q 1.2 | Page 48

Use suitable identities to find the following products :- (x + 8) (x – 10)

Exercise 2.5 | Q 1.3 | Page 48

Use suitable identities to find the following products :- (3x + 4) (3x – 5)

Exercise 2.5 | Q 1.4 | Page 48

Use suitable identities to find the following products :-

`(y^2+3/2)(y^2-3/2)`

 

Exercise 2.5 | Q 1.5 | Page 48

Use suitable identities to find the following products :- (3 – 2x) (3 + 2x)

Exercise 2.5 | Q 2.1 | Page 48

Evaluate the following product without multiplying directly:- 103 × 107

Exercise 2.5 | Q 2.2 | Page 48

Evaluate the following product without multiplying directly:- 95 × 96

Exercise 2.5 | Q 2.3 | Page 48

Evaluate the following product without multiplying directly:- 104 × 96

Exercise 2.5 | Q 3.1 | Page 48

Factorise the following using appropriate identities:- 9x2 + 6xy + y2 

Exercise 2.5 | Q 3.2 | Page 48

Factorise the following using appropriate identities:- 4y2 – 4y + 1

Exercise 2.5 | Q 3.3 | Page 48

Factorise the following using appropriate identities:- `x^2 - y^2/100`

 

Exercise 2.5 | Q 4.1 | Page 49

Expand following, using suitable identities :- (x + 2y + 4z)

Exercise 2.5 | Q 4.2 | Page 49

Expand following, using suitable identities :- (2x – y + z)2 

Exercise 2.5 | Q 4.3 | Page 49

Expand following, using suitable identities :- (–2x + 3y + 2z)2 

 

Exercise 2.5 | Q 4.4 | Page 49

Expand following, using suitable identities :- (3a – 7b – c)2 

Exercise 2.5 | Q 4.5 | Page 49

Expand following, using suitable identities :- (–2x + 5y – 3z)2 

Exercise 2.5 | Q 4.6 | Page 49

Expand following, using suitable identities :-`[1/4a-1/2b+1]^2`

 

Exercise 2.5 | Q 5.1 | Page 49

Factorise:- 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz

Exercise 2.5 | Q 5.2 | Page 49

Factorise:- `2x^2+y^2+8z^2-2sqrt2xy+4sqrt2yz-8xz`

 

Exercise 2.5 | Q 6.1 | Page 49

Write the following cube in expanded form:- (2x + 1)3 

Exercise 2.5 | Q 6.2 | Page 49

Write the following cube in expanded form:- (2a – 3b)3 

Exercise 2.5 | Q 6.3 | Page 49

Write the following cube in expanded form:- `[3/2x+1]^3`

 

Exercise 2.5 | Q 6.4 | Page 49

Write the following cube in expanded form:- `[x-2/3y]^3`

 

Exercise 2.5 | Q 7.1 | Page 49

Evaluate the following using suitable identities:-

(99)3 

Exercise 2.5 | Q 7.2 | Page 49

Evaluate the following using suitable identities:- (102)3 

Exercise 2.5 | Q 7.3 | Page 49

Evaluate the following using suitable identities:- (998)3 

Exercise 2.5 | Q 8.1 | Page 49

Factorise :- 8a3 + b3 + 12a2b + 6ab2 

Exercise 2.5 | Q 8.2 | Page 49

Factorise :- 8a3 – b3 – 12a2b + 6ab2

Exercise 2.5 | Q 8.3 | Page 49

Factorise :- 27 – 125a3 – 135a + 225a2 

Exercise 2.5 | Q 8.4 | Page 49

Factorise :- 64a3 – 27b3 – 144a2b + 108ab2 

Exercise 2.5 | Q 8.5 | Page 49

Factorise :-

`27p^3-1/216-9/2p^2+1/4p`

 

Exercise 2.5 | Q 9.1 | Page 49

Verify :-  x3 + y3 = (x + y) (x2 – xy + y2)

Exercise 2.5 | Q 9.2 | Page 49

Verify :- x3 – y3 = (x – y) (x2 + xy + y2)

Exercise 2.5 | Q 10.1 | Page 49

Factorise :- 27y3 + 125z3

Exercise 2.5 | Q 10.2 | Page 49

Factorise :- 64m3 – 343n3 

Exercise 2.5 | Q 11 | Page 49

Factorise :- 27x3 + y3 + z3 – 9xyz

Exercise 2.5 | Q 12 | Page 49

Verify that `x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`

 

Exercise 2.5 | Q 13 | Page 49

If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.

Exercise 2.5 | Q 14.1 | Page 49

Without actually calculating the cubes, find the value of the following:- (–12)3 + (7)3 + (5)3 

Exercise 2.5 | Q 14.2 | Page 49

Without actually calculating the cubes, find the value of the following:- (28)3 + (–15)3 + (–13)3

Exercise 2.5 | Q 15.1 | Page 49

Give possible expressions for the length and breadth of the following rectangle, in which their areas are given:-

Area : 25a2 – 35a + 12
Exercise 2.5 | Q 15.2 | Page 49

Give possible expressions for the length and breadth of the following rectangle, in which their areas are given:-

Area : 35y2 + 13y –12
Exercise 2.5 | Q 16.1 | Page 50

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

Volume : 3x2 – 12x
Exercise 2.5 | Q 16.2 | Page 50

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

Volume : 12ky2 + 8ky – 20k

Solutions for Chapter 2: Polynomials

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5

NCERT solutions for Mathematics Class 9 chapter 2 - Polynomials

Shaalaa.com has the CBSE Mathematics Mathematics Class 9 CBSE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Mathematics Class 9 CBSE 2 (Polynomials) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Mathematics Class 9 chapter 2 Polynomials are Algebraic Identities, Algebraic Identities, Polynomials, Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials, Factorising the Quadratic Polynomial (Trinomial) of the type ax2 + bx + c, a ≠ 0., Algebraic Identities, Polynomials, Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials, Factorising the Quadratic Polynomial (Trinomial) of the type ax2 + bx + c, a ≠ 0..

Using NCERT Mathematics Class 9 solutions Polynomials exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE Mathematics Class 9 students prefer NCERT Textbook Solutions to score more in exams.

Get the free view of Chapter 2, Polynomials Mathematics Class 9 additional questions for Mathematics Mathematics Class 9 CBSE, and you can use Shaalaa.com to keep it handy for your exam preparation.

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