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# NCERT solutions for Class 9 Mathematics chapter 2 - Polynomials

## Mathematics Textbook for Class 9

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#### NCERT Mathematics Class 9 ## Chapter 2: Polynomials

Ex. 21OthersEx. 2.1Ex. 2.2Ex. 2.3Ex. 2.4Ex. 2.5

#### Chapter 2: Polynomials Exercise 2.1, 21 solutions [Page 32]

Q 1 | Page 32

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

(i) 4x^2 - 3x + 7

(ii) y^2+sqrt2

(iii) 3sqrtt+tsqrt2

(iv) y+2/y

(v) x^10+y^3+t^50

Ex. 2.1 | Q 2.1 | Page 32

Write the coefficient of x2 in the following:-

2+x^2+x

Ex. 2.1 | Q 2.2 | Page 32

Write the coefficient of x2 in the following:-

2-x^2+x^3

Ex. 2.1 | Q 2.3 | Page 32

Write the coefficient of x2 in the following:-

pi/2x^2+x

Ex. 2.1 | Q 2.4 | Page 32

Write the coefficient of x2 in the following:-

sqrt2x-1

Ex. 21 | Q 3 | Page 32

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

Ex. 2.1 | Q 4.1 | Page 32

Write the degree of the following polynomial:-

5x3 + 4x2 +7x

Ex. 2.1 | Q 4.2 | Page 32

Write the degree of the following polynomial:-

4 - y2

Ex. 2.1 | Q 4.3 | Page 32

Write the degree of the following polynomials:-

5t - sqrt7

Ex. 2.1 | Q 4.4 | Page 32

Write the degree of each of the following polynomials:-

3

Ex. 2.1 | Q 5 | Page 32

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

(i) x2 + x

(ii) x – x3

(iii) y + y2 + 4

(iv) 1 + x

(v) 3t

(vi) r2

(vii) 7x3

#### Chapter 2: Polynomials Exercise 2.2 solutions [Pages 34 - 35]

Ex. 2.2 | Q 1.1 | Page 34

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

x = 0

Ex. 2.2 | Q 1.2 | Page 34

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

x = –1

Ex. 2.2 | Q 1.3 | Page 34

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

x = 2

Ex. 2.2 | Q 2.1 | Page 34

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

p(y) = y2 – y + 1

Ex. 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

Ex. 2.2 | Q 2.3 | Page 34

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

p(x) = x3

Ex. 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)

Ex. 2.2 | Q 3.1 | Page 35

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

p(x) = 3x + 1, x =-1/3

Ex. 2.2 | Q 3.2 | Page 35

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

p(x) = 5x – π, x = 4/5

Ex. 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

Ex. 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

Ex. 2.2 | Q 3.5 | Page 35

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

p(x) = x2, x = 0

Ex. 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

Ex. 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

Ex. 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

Ex. 2.2 | Q 4.1 | Page 35

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

Ex. 2.2 | Q 4.2 | Page 35

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

Ex. 2.2 | Q 4.3 | Page 35

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

Ex. 2.2 | Q 4.4 | Page 35

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

Ex. 2.2 | Q 4.5 | Page 35

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

Ex. 2.2 | Q 4.6 | Page 35

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

p(x) = ax, a ≠ 0

Ex. 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.

#### Chapter 2: Polynomials Exercise 2.3 solutions [Page 40]

Ex. 2.3 | Q 1.1 | Page 40

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

Ex. 2.3 | Q 1.2 | Page 40

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

Ex. 2.3 | Q 1.3 | Page 40

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

Ex. 2.3 | Q 1.4 | Page 40

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

Ex. 2.3 | Q 1.5 | Page 40

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

Ex. 2.3 | Q 2 | Page 40

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

Ex. 2.3 | Q 3 | Page 40

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

#### Chapter 2: Polynomials Exercise 2.4 solutions [Pages 43 - 44]

Ex. 2.4 | Q 1.1 | Page 43

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

x3 + x2 + x + 1

Ex. 2.4 | Q 1.2 | Page 43

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

x4 + x3 + x2 + x + 1

Ex. 2.4 | Q 1.3 | Page 43

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

x4 + 3x3 + 3x2 + x + 1

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 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

Ex. 2.4 | Q 4.1 | Page 44

Factorise :- 12x2 – 7x + 1

Ex. 2.4 | Q 4.2 | Page 44

Factorise :- 2x2 + 7x + 3

Ex. 2.4 | Q 4.3 | Page 44

Factorise :- 6x2 + 5x – 6

Ex. 2.4 | Q 4.4 | Page 44

Factorise :- 3x2 – x – 4

Ex. 2.4 | Q 5.1 | Page 44

Factorise :- x3 – 2x2 – x + 2

Ex. 2.4 | Q 5.2 | Page 44

Factorise :- x3 – 3x2 – 9x – 5

Ex. 2.4 | Q 5.3 | Page 44

Factorise :- x3 + 13x2 + 32x + 20

Ex. 2.4 | Q 5.4 | Page 44

Factorise :- 2y3 + y2 – 2y – 1

#### Chapter 2: Polynomials Exercise 2.5 solutions [Pages 48 - 50]

Ex. 2.5 | Q 1.1 | Page 48

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

Ex. 2.5 | Q 1.2 | Page 48

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

Ex. 2.5 | Q 1.3 | Page 48

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

Ex. 2.5 | Q 1.4 | Page 48

Use suitable identities to find the following products :-

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

Ex. 2.5 | Q 1.5 | Page 48

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

Ex. 2.5 | Q 2.1 | Page 48

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

Ex. 2.5 | Q 2.2 | Page 48

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

Ex. 2.5 | Q 2.3 | Page 48

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

Ex. 2.5 | Q 3.1 | Page 48

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

Ex. 2.5 | Q 3.2 | Page 48

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

Ex. 2.5 | Q 3.3 | Page 48

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

Ex. 2.5 | Q 4.1 | Page 49

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

Ex. 2.5 | Q 4.2 | Page 49

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

Ex. 2.5 | Q 4.3 | Page 49

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

Ex. 2.5 | Q 4.4 | Page 49

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

Ex. 2.5 | Q 4.5 | Page 49

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

Ex. 2.5 | Q 4.6 | Page 49

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

Ex. 2.5 | Q 5.1 | Page 49

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

Ex. 2.5 | Q 5.2 | Page 49

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

Ex. 2.5 | Q 6.1 | Page 49

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

Ex. 2.5 | Q 6.2 | Page 49

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

Ex. 2.5 | Q 6.3 | Page 49

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

Ex. 2.5 | Q 6.4 | Page 49

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

Ex. 2.5 | Q 7.1 | Page 49

Evaluate the following using suitable identities:-

(99)3

Ex. 2.5 | Q 7.2 | Page 49

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

Ex. 2.5 | Q 7.3 | Page 49

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

Ex. 2.5 | Q 8.1 | Page 49

Factorise :- 8a3 + b3 + 12a2b + 6ab2

Ex. 2.5 | Q 8.2 | Page 49

Factorise :- 8a3 – b3 – 12a2b + 6ab2

Ex. 2.5 | Q 8.3 | Page 49

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

Ex. 2.5 | Q 8.4 | Page 49

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

Ex. 2.5 | Q 8.5 | Page 49

Factorise :-

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

Ex. 2.5 | Q 9.1 | Page 49

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

Ex. 2.5 | Q 9.2 | Page 49

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

Ex. 2.5 | Q 10.1 | Page 49

Factorise :- 27y3 + 125z3

Ex. 2.5 | Q 10.2 | Page 49

Factorise :- 64m3 – 343n3

Ex. 2.5 | Q 11 | Page 49

Factorise :- 27x3 + y3 + z3 – 9xyz

Ex. 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]

Ex. 2.5 | Q 13 | Page 49

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

Ex. 2.5 | Q 14.1 | Page 49

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

Ex. 2.5 | Q 14.2 | Page 49

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

Ex. 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
Ex. 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
Ex. 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
Ex. 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

## Chapter 2: Polynomials

Ex. 21OthersEx. 2.1Ex. 2.2Ex. 2.3Ex. 2.4Ex. 2.5

#### NCERT Mathematics Class 9 ## NCERT solutions for Class 9 Mathematics chapter 2 - Polynomials

NCERT solutions for Class 9 Maths chapter 2 (Polynomials) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Textbook for Class 9 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 9 Mathematics chapter 2 Polynomials are Algebraic Identities, Introduction of Polynomials, Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials.

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

Get the free view of chapter 2 Polynomials Class 9 extra questions for Maths and can use shaalaa.com to keep it handy for your exam preparation

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