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

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#### Chapters ## Chapter 2: Polynomials

OthersEx. 2.1Ex. 21.00Ex. 2.2Ex. 2.3Ex. 2.4Ex. 2.5

#### NCERT solutions for Class 9 Mathematics Textbook Chapter 2 Polynomials Exercise 2.1, 21.00 [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. 21.00 | 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

#### NCERT solutions for Class 9 Mathematics Textbook Chapter 2 Polynomials Exercise 2.2 [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.

#### NCERT solutions for Class 9 Mathematics Textbook Chapter 2 Polynomials Exercise 2.3 [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.

#### NCERT solutions for Class 9 Mathematics Textbook Chapter 2 Polynomials Exercise 2.4 [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

#### NCERT solutions for Class 9 Mathematics Textbook Chapter 2 Polynomials Exercise 2.5 [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

OthersEx. 2.1Ex. 21.00Ex. 2.2Ex. 2.3Ex. 2.4Ex. 2.5 ## NCERT solutions for Class 9 Mathematics Textbook chapter 2 - Polynomials

NCERT solutions for Class 9 Mathematics Textbook 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 Class 9 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide 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 Textbook 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.

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