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

## Chapter 2: Polynomials

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4
Exercise 2.1 [Pages 14 - 16]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 2 Polynomials Exercise 2.1 [Pages 14 - 16]

#### Write the correct answer in the following:

Exercise 2.1 | Q 1 | Page 14

Which one of the following is a polynomial?

• x^2/2 - 2/x^2

• sqrt(2x) - 1

• x^2 + (3x^(3/2))/sqrt(x)

• (x - 1)/(x + 1)

Exercise 2.1 | Q 2 | Page 14

sqrt(2) is a polynomial of degree ______.

• 2

• 0

• 1

• 1/2

Exercise 2.1 | Q 3 | Page 14

Degree of the polynomial 4x4 + 0x3 + 0x5 + 5x + 7 is ______.

• 4

• 5

• 3

• 7

Exercise 2.1 | Q 4 | Page 14

Degree of the zero polynomial is ______.

• 0

• 1

• Any natural number

• Not defined

Exercise 2.1 | Q 5 | Page 14

If p(x) = x^2 - 2sqrt(2)x + 1, then p(2sqrt(2)) is equal to ______.

• 0

• 1

• 4sqrt(2)

• 8sqrt(2) + 1

Exercise 2.1 | Q 6 | Page 14

The value of the polynomial 5x – 4x2 + 3, when x = –1 is ______.

• – 6

• 6

• 2

• –2

Exercise 2.1 | Q 7 | Page 15

If p(x) = x + 3, then p(x) + p(–x) is equal to ______.

• 3

• 2x

• 0

• 6

Exercise 2.1 | Q 8 | Page 15

Zero of the zero polynomial is ______.

• 0

• 1

• Any real number

• Not defined

Exercise 2.1 | Q 9 | Page 15

Zero of the polynomial p(x) = 2x + 5 is ______.

• -2/5

• -5/2

• 2/5

• 5/2

Exercise 2.1 | Q 10 | Page 15

One of the zeroes of the polynomial 2x^2 + 7x - 4 is ______.

• 2

• 1/2

• -1/2

• -2

Exercise 2.1 | Q 11 | Page 15

If x51 + 51 is divided by x + 1, the remainder is ______.

• 0

• 1

• 49

• 50

Exercise 2.1 | Q 12 | Page 15

If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is ______.

• –3

• 4

• 2

• –2

Exercise 2.1 | Q 13 | Page 15

x + 1 is a factor of the polynomial ______.

• x3 + x2 – x + 1

• x3 + x2 + x + 1

• x4 + x3 + x2 +1

• x4 + 3x3 + 3x2 + x + 1

Exercise 2.1 | Q 14 | Page 15

One of the factors of (25x2 – 1) + (1 + 5x)2 is ______.

• 5 + x

• 5 – x

• 5x – 1

• 10x

Exercise 2.1 | Q 15 | Page 15

The value of 2492 – 2482 is ______.

• 12

• 477

• 487

• 497

Exercise 2.1 | Q 16 | Page 15

The factorisation of 4x2 + 8x + 3 is ______.

• (x + 1)(x + 3)

• (2x + 1)(2x + 3)

• (2x + 2)(2x + 5)

• (2x –1)(2x –3)

Exercise 2.1 | Q 17 | Page 15

Which of the following is a factor of (x + y)3 – (x3 + y3)?

• x2 + y2 + 2xy

• x2 + y2 – xy

• xy2

• 3xy

Exercise 2.1 | Q 18 | Page 15

The coefficient of x in the expansion of (x + 3)3 is ______.

• 1

• 9

• 18

• 27

Exercise 2.1 | Q 19 | Page 15

If x/y + y/x = -1 (x, y ≠ 0), the value of x3 – y3 is ______.

• 1

• –1

• 0

• 1/2

Exercise 2.1 | Q 20 | Page 16

If 49x^2 - b = (7x + 1/2)(7x - 1/2), then the value of b is ______.

• 0

• 1/sqrt(2)

• 1/4

• 1/2

Exercise 2.1 | Q 21 | Page 16

If a + b + c = 0, then a3 + b3 + c3 is equal to ______.

• 0

• abc

• 3abc

• 2abc

Exercise 2.2 [Pages 16 - 17]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 2 Polynomials Exercise 2.2 [Pages 16 - 17]

Exercise 2.2 | Q 1.(i) | Page 16

8

Exercise 2.2 | Q 1.(ii) | Page 16

sqrt(3)x^2 - 2x

Exercise 2.2 | Q 1.(iii) | Page 16

1 - sqrt(5)x

Exercise 2.2 | Q 1.(iv) | Page 16

1/(5x^-2) + 5x + 7

Exercise 2.2 | Q 1.(v) | Page 16

((x - 2)(x - 4))/x

Exercise 2.2 | Q 1.(vi) | Page 16

1/(x + 1)

Exercise 2.2 | Q 1.(vii) | Page 16

1/7 a^3 - 2/sqrt(3) a^2 + 4a - 7

Exercise 2.2 | Q 1.(viii) | Page 16

1/(x + 1)

Exercise 2.2 | Q 1.(viii) | Page 16

1/(x + 1)

#### State whether the following statement is True or False:

Exercise 2.2 | Q 2.(i) | Page 17

A binomial can have atmost two terms

• True

• False

Exercise 2.2 | Q 2.(ii) | Page 17

Every polynomial is a Binomial.

• True

• False

Exercise 2.2 | Q 2.(iii) | Page 17

A binomial may have degree 5

• True

• False

Exercise 2.2 | Q 2.(iv) | Page 17

Zero of a polynomial is always 0

• True

• False

Exercise 2.2 | Q 2.(v) | Page 17

A polynomial cannot have more than one zero

• True

• False

Exercise 2.2 | Q 2.(vi) | Page 17

The degree of the sum of two polynomials each of degree 5 is always 5.

• True

• False

Exercise 2.3 [Pages 18 - 22]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 2 Polynomials Exercise 2.3 [Pages 18 - 22]

Exercise 2.3 | Q 1.(i) | Page 18

Classify the following polynomial as polynomials in one variable, two variables etc.

x2 + x + 1

Exercise 2.3 | Q 1.(ii) | Page 18

Classify the following polynomial as polynomials in one variable, two variables etc.

y3 – 5y

Exercise 2.3 | Q 1.(iii) | Page 18

Classify the following polynomial as polynomials in one variable, two variables etc.

xy + yz + zx

Exercise 2.3 | Q 1.(iv) | Page 18

Classify the following polynomial as polynomials in one variable, two variables etc.

x2 – 2xy + y2 + 1

Exercise 2.3 | Q 2.(i) | Page 19

Determine the degree of the following polynomials:

2x – 1

Exercise 2.3 | Q 2.(ii) | Page 19

Determine the degree of the following polynomials:

–10

Exercise 2.3 | Q 2.(iii) | Page 19

Determine the degree of the following polynomials:

x3 – 9x + 3x5

Exercise 2.3 | Q 2.(iv) | Page 19

Determine the degree of the following polynomials:

y3(1 – y4)

Exercise 2.3 | Q 3.(i) | Page 19

For the polynomial ((x^3 + 2x + 1))/5 - 7/2 x^2 - x^6, write the degree of the polynomial

Exercise 2.3 | Q 3.(ii) | Page 19

For the polynomial ((x^3 + 2x + 1))/5 - 7/2 x^2 - x^6, write the coefficient of x3

Exercise 2.3 | Q 3.(iii) | Page 19

For the polynomial ((x^3 + 2x + 1))/5 - 7/2 x^2 - x^6, write the coefficient of x6

Exercise 2.3 | Q 3.(iv) | Page 19

For the polynomial ((x^3 + 2x + 1))/5 - 7/2 x^2 - x^6, write the constant term

Exercise 2.3 | Q 4.(i) | Page 19

Write the coefficient of x2 in the following:

pi/6 x + x^2 - 1

Exercise 2.3 | Q 4.(ii) | Page 19

Write the coefficient of x2 in the following:

3x – 5

Exercise 2.3 | Q 4.(iii) | Page 19

Write the coefficient of x2 in the following:

(x – 1)(3x – 4)

Exercise 2.3 | Q 4.(iv) | Page 19

Write the coefficient of x2 in the following:

(2x – 5)(2x2 – 3x + 1)

Exercise 2.3 | Q 5.(i) | Page 19

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

2 – x2 + x3

Exercise 2.3 | Q 5.(ii) | Page 19

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

3x3

Exercise 2.3 | Q 5.(iii) | Page 19

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

5t - sqrt(7)

Exercise 2.3 | Q 5.(iv) | Page 19

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

4 – 5y2

Exercise 2.3 | Q 5.(v) | Page 19

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

3

Exercise 2.3 | Q 5.(vi) | Page 19

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

2 + x

Exercise 2.3 | Q 5.(vii) | Page 19

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

y3 – y

Exercise 2.3 | Q 5.(viii) | Page 19

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

1 + x + x2

Exercise 2.3 | Q 5.(ix) | Page 19

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

t2

Exercise 2.3 | Q 5.(x) | Page 19

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

sqrt(2)x - 1

Exercise 2.3 | Q 6.(i) | Page 19

Give an example of a polynomial, which is monomial of degree 1

Exercise 2.3 | Q 6.(ii) | Page 19

Give an example of a polynomial, which is binomial of degree 20

Exercise 2.3 | Q 6.(iii) | Page 19

Give an example of a polynomial, which is trinomial of degree 2

Exercise 2.3 | Q 7 | Page 19

Find the value of the polynomial 3x3 – 4x2 + 7x – 5, when x = 3 and also when x = –3.

Exercise 2.3 | Q 8 | Page 19

If p(x) = x2 – 4x + 3, then evaluate p(2) – p(–1) + p(1/2).

Exercise 2.3 | Q 9.(i) | Page 19

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

p(x) = 10x – 4x2 – 3

Exercise 2.3 | Q 9.(ii) | Page 19

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

p(y) = (y + 2)(y - 2)

#### Verify whether the following are True or False:

Exercise 2.3 | Q 10.(i) | Page 19

–3 is a zero of x – 3

• True

• False

Exercise 2.3 | Q 10.(ii) | Page 19

-1/3 is a zero of 3x + 1

• True

• False

Exercise 2.3 | Q 10.(iii) | Page 19

(-4)/5 is a zero of 4 – 5y

• True

• False

Exercise 2.3 | Q 10.(iv) | Page 19

0 and 2 are the zeroes of t2 – 2t

• True

• False

Exercise 2.3 | Q 10.(v) | Page 19

–3 is a zero of y2 + y – 6

• True

• False

Exercise 2.3 | Q 11.(i) | Page 20

Find the zeroes of the polynomial in the following:

p(x) = x – 4

Exercise 2.3 | Q 11.(ii) | Page 20

Find the zeroes of the polynomial in the following:

g(x) = 3 – 6x

Exercise 2.3 | Q 11.(iii) | Page 20

Find the zeroes of the polynomial in the following:

q(x) = 2x – 7

Exercise 2.3 | Q 11.(iv) | Page 20

Find the zeroes of the polynomial in the following:

h(y) = 2y

Exercise 2.3 | Q 12 | Page 20

Find the zeroes of the polynomial:

p(x) = (x – 2)2 – (x + 2)2

Exercise 2.3 | Q 13 | Page 20

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: x4 + 1; x – 1

Exercise 2.3 | Q 14.(i) | Page 20

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where p(x) = x3 – 2x2 – 4x – 1, g(x) = x + 1

Exercise 2.3 | Q 14.(ii) | Page 20

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where p(x) = x3 – 3x2 + 4x + 50, g(x) = x – 3

Exercise 2.3 | Q 14.(iii) | Page 20

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where p(x) = 4x3 – 12x2 + 14x – 3, g(x) = 2x – 1

Exercise 2.3 | Q 14.(iv) | Page 20

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where p(x) = x3 – 6x2 + 2x – 4, g(x) = 1 - 3/2 x

Exercise 2.3 | Q 15.(i) | Page 20

Check whether p(x) is a multiple of g(x) or not:
p(x) = x3 – 5x2 + 4x – 3, g(x) = x – 2

Exercise 2.3 | Q 15.(ii) | Page 20

Check whether p(x) is a multiple of g(x) or not:
p(x) = 2x3 – 11x2 – 4x + 5, g(x) = 2x + 1

Exercise 2.3 | Q 16.(i) | Page 20

Show that: x + 3 is a factor of 69 + 11x – x2 + x3.

Exercise 2.3 | Q 16.(ii) | Page 20

Show that: 2x – 3 is a factor of x + 2x3 – 9x2 + 12.

Exercise 2.3 | Q 17.(i) | Page 20

Determine which of the following polynomials has x – 2 a factor:

3x2 + 6x – 24

Exercise 2.3 | Q 17.(ii) | Page 20

Determine which of the following polynomials has x – 2 a factor:

4x2 + x – 2

Exercise 2.3 | Q 17.(ii) | Page 20

Determine which of the following polynomials has x – 2 a factor:

4x2 + x – 2

Exercise 2.3 | Q 18 | Page 20

Show that p – 1 is a factor of p10 – 1 and also of p11 – 1.

Exercise 2.3 | Q 19 | Page 20

For what value of m is x3 – 2mx2 + 16 divisible by x + 2?

Exercise 2.3 | Q 20 | Page 20

If x + 2a is a factor of x5 – 4a2x3 + 2x + 2a + 3, find a.

Exercise 2.3 | Q 21 | Page 20

Find the value of m so that 2x – 1 be a factor of 8x4 + 4x3 – 16x2 + 10x + m.

Exercise 2.3 | Q 22 | Page 21

If x + 1 is a factor of ax3 + x2 – 2x + 4a – 9, find the value of a.

Exercise 2.3 | Q 23.(i) | Page 21

Factorise: x2 + 9x + 18

Exercise 2.3 | Q 23.(ii) | Page 21

Factorise: 6x2 + 7x – 3

Exercise 2.3 | Q 23.(iii) | Page 21

Factorise: 2x2 – 7x – 15

Exercise 2.3 | Q 23.(iv) | Page 21

Factorise: 84 – 2r – 2r2

Exercise 2.3 | Q 24.(i) | Page 21

Factorise: 2x3 – 3x2 – 17x + 30

Exercise 2.3 | Q 24.(ii) | Page 21

Factorise: x3 – 6x2 + 11x – 6

Exercise 2.3 | Q 24.(iii) | Page 21

Factorise: x3 + x2 – 4x – 4

Exercise 2.3 | Q 24.(iv) | Page 21

Factorise: 3x3 – x2 – 3x + 1

Exercise 2.3 | Q 25.(i) | Page 21

Using suitable identity, evaluate the following:
1033

Exercise 2.3 | Q 25.(ii) | Page 21

Using suitable identity, evaluate the following:
101 × 102

Exercise 2.3 | Q 25.(iii) | Page 21

Using suitable identity, evaluate the following:
9992

Exercise 2.3 | Q 26.(i) | Page 21

Factorise the following:

4x^2 + 20x + 25

Exercise 2.3 | Q 26.(ii) | Page 21

Factorise the following:

9y^2 - 66yz + 121z^2

Exercise 2.3 | Q 26.(iii) | Page 21

Factorise the following:

(2x + 1/3)^2 - (x - 1/2)^2

Exercise 2.3 | Q 27.(i) | Page 21

Factorise the following :

9x2 – 12x + 3

Exercise 2.3 | Q 27.(ii) | Page 21

Factorise the following:

9x2 – 12x + 4

Exercise 2.3 | Q 28.(i) | Page 21

Expand the following:

(4a - b + 2c)^2

Exercise 2.3 | Q 28.(ii) | Page 21

Expand the following:

(3a - 5b - c)^2

Exercise 2.3 | Q 28.(iii) | Page 21

Expand the following:

(-x + 2y - 3z)^2

Exercise 2.3 | Q 29.(i) | Page 21

Factorise the following:

9x2 + 4y2 + 16z2 + 12xy – 16yz – 24xz

Exercise 2.3 | Q 29.(ii) | Page 21

Factorise the following:

25x2 + 16y2 + 4z2 – 40xy + 16yz – 20xz

Exercise 2.3 | Q 29.(iii) | Page 21

Factorise the following:

16x2 + 4y2 + 9z2 – 16xy – 12yz + 24xz

Exercise 2.3 | Q 30 | Page 21

If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 + c2.

Exercise 2.3 | Q 31.(i) | Page 21

Expand the following:
(3a – 2b)3

Exercise 2.3 | Q 31.(ii) | Page 21

Expand the following:
(1/x + y/3)^3

Exercise 2.3 | Q 31.(iii) | Page 21

Expand the following:
(4 - 1/(3x))^3

Exercise 2.3 | Q 32.(i) | Page 21

Factorise the following:

1 - 64a^3 - 12a + 48a^2

Exercise 2.3 | Q 32.(ii) | Page 21

Factorise the following:

8p^3 + 12/5 p^2 + 6/25 p + 1/125

Exercise 2.3 | Q 33.(i) | Page 22

Find the following products:

(x/2 + 2y)(x^2/4 - xy  4y^2)

Exercise 2.3 | Q 33.(ii) | Page 22

Find the following products:

(x^2 - 1)(x^4 + x^2 + 1)

Exercise 2.3 | Q 34.(i) | Page 22

Factorise: 1 + 64x3

Exercise 2.3 | Q 34.(ii) | Page 22

Factorise: a^3 - 2sqrt(2)b^3

Exercise 2.3 | Q 35 | Page 22

Find the following product:

(2x - y + 3z)(4x^2 + y^2 + 9z^2 + 2xy + 3yz - 6xz)

Exercise 2.3 | Q 36.(i) | Page 22

Factorise: a^3 - 8b^3 - 64c^3 - 24abc

Exercise 2.3 | Q 36.(ii) | Page 22

Factorise: 2sqrt(2)a^3 + 8b^3 - 27c^3 + 18sqrt(2)abc

Exercise 2.3 | Q 37.(i) | Page 22

Without actually calculating the cubes, find the value of:

(1/2)^3 + (1/3)^3 - (5/6)^3

Exercise 2.3 | Q 37.(ii) | Page 22

Without actually calculating the cubes, find the value of:

(0.2)^3 - (0.3)^3 + (0.1)^3

Exercise 2.3 | Q 38 | Page 22

Without finding the cubes, factorise:

(x - 2y)^3 + (2y - 3z)^3 + (3z - x)^3

Exercise 2.3 | Q 39.(i) | Page 22

Find the value of x3 + y3 – 12xy + 64,when x + y = – 4.

Exercise 2.3 | Q 39.(ii) | Page 22

Find the value of x3 – 8y3 – 36xy – 216, when x = 2y + 6

Exercise 2.3 | Q 40 | Page 22

Give possible expressions for the length and breadth of the rectangle whose area is given by 4a2 + 4a – 3.

Exercise 2.4 [Page 23]

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 2 Polynomials Exercise 2.4 [Page 23]

Exercise 2.4 | Q 1 | Page 23

If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + o leave the same remainder when divided by z – 3, find the value of a.

Exercise 2.4 | Q 2 | Page 23

The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the values of a. Also find the remainder when p(x) is divided by x + 2.

Exercise 2.4 | Q 3 | Page 23

If both x - 2 and x - 1/2 are factors of px^2 + 5x + r, show that p = r.

Exercise 2.4 | Q 4 | Page 23

Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2.

Exercise 2.4 | Q 5 | Page 23

Simplify: (2x – 5y)3 – (2x + 5y)3.

Exercise 2.4 | Q 6 | Page 23

Multiply x2 + 4y2 + z2 + 2xy + xz – 2yz by (– z + x – 2y).

Exercise 2.4 | Q 7 | Page 23

If a, b, c are all non-zero and a + b + c = 0, prove that a^2/(bc) + b^2/(ca) + c^2/(ab) = 3.

Exercise 2.4 | Q 8 | Page 23

If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 – 3abc = – 25.

Exercise 2.4 | Q 9 | Page 23

Prove that (a + b + c)3 – a3 – b– c3 = 3(a + b)(b + c)(c + a).

## Chapter 2: Polynomials

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4

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

NCERT solutions for Mathematics Exemplar Class 9 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 Exemplar Class 9 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 Mathematics Exemplar Class 9 chapter 2 Polynomials are 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 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 Mathematics Exemplar Class 9 and can use Shaalaa.com to keep it handy for your exam preparation