#### Chapters

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equations in two Variables

Chapter 5: Introduction to Euclid's Geometry

Chapter 6: Lines and Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Areas of Parallelograms and Triangles

Chapter 10: Circles

Chapter 11: Constructions

Chapter 12: Heron's Formula

Chapter 13: Surface Area and Volumes

Chapter 14: Statistics

Chapter 15: Probability

#### NCERT Mathematics Class 9

## Chapter 2: Polynomials

#### Chapter 2: Polynomials Exercise 2.1, 21 solutions [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`

Write the coefficient of x^{2} in the following:-

`2+x^2+x`

Write the coefficient of x^{2} in the following:-

`2-x^2+x^3`

Write the coefficient of x^{2} in the following:-

`pi/2x^2+x`

Write the coefficient of x^{2} in the following:-

`sqrt2x-1`

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

Write the degree of the following polynomial:-

5x^{3} + 4x^{2} +7x

Write the degree of the following polynomial:-

4 - y^{2}

Write the degree of the following polynomials:-

`5t - sqrt7`

Write the degree of each of the following polynomials:-

3

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

(i) x^{2} + x

(ii) x – x^{3}

(iii) y + y^{2} + 4

(iv) 1 + x

(v) 3t

(vi) r^{2}

(vii) 7x^{3}

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

Find the value of the polynomial 5x – 4x^{2} + 3 at

x = 0

Find the value of the polynomial 5x – 4x^{2} + 3 at

x = –1

Find the value of the polynomial 5x – 4x^{2} + 3 at

x = 2

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

p(y) = y^{2} – y + 1

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

p(t) = 2 + t + 2t^{2} – t^{3}

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

p(x) = x^{3}

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

p(x) = (x – 1) (x + 1)

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

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

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

p(x) = 5x – π, x = 4/5

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

p(x) = x^{2} – 1, x = 1, –1

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

p(x) = (x + 1) (x – 2), x = – 1, 2

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

p(x) = x^{2}, x = 0

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

p(x) = lx + m, `x = – m/l`

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

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

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

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

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

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

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

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

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

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

p(x) = ax, a ≠ 0

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]

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by x+1.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by `x - 1/2`

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by x.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by x + π.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by 5 + 2x.

Find the remainder when x^{3} – ax^{2} + 6x – a is divided by x – a.

Check whether 7 + 3x is a factor of 3x^{3} + 7x.

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

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

x^{3} + x^{2} + x + 1

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

x^{4} + x^{3} + x^{2} + x + 1

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

x^{4} + 3x^{3} + 3x^{2} + x + 1

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

`x^3-x^2-(2+sqrt2)x+sqrt2`

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

p(x) = 2x^{3} + x^{2} – 2x – 1, g(x) = x + 1

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

*p*(*x*) = *x*^{3} + 3*x*^{2} + 3*x* + 1, *g*(*x*) = *x* + 2

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

*p*(*x*) = *x*^{3} − 4*x*^{2} + *x* + 6, *g*(*x*) = *x* − 3

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

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

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

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

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

p(x) = kx^{2} – 3x + k

Factorise :- 12x^{2} – 7x + 1

Factorise :- 2x^{2} + 7x + 3

Factorise :- 6x^{2} + 5x – 6

Factorise :- 3x^{2} – x – 4

Factorise :- x^{3} – 2x^{2} – x + 2

Factorise :- x^{3} – 3x^{2} – 9x – 5

Factorise :- x^{3} + 13x^{2} + 32x + 20

Factorise :- 2y^{3} + y^{2} – 2y – 1

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

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

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

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

Use suitable identities to find the following products :-

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

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

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

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

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

Factorise the following using appropriate identities:- 9x^{2} + 6xy + y^{2}

Factorise the following using appropriate identities:- 4y^{2} – 4y + 1

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

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

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

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

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

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

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

Factorise:- 4x^{2} + 9y^{2} + 16z^{2} + 12xy – 24yz – 16xz

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

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

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

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

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

Evaluate the following using suitable identities:-

(99)^{3}

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

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

Factorise :- 8a^{3} + b^{3} + 12a^{2}b + 6ab^{2}

Factorise :- 8a^{3} – b^{3} – 12a^{2}b + 6ab^{2}

Factorise :- 27 – 125a^{3} – 135a + 225a^{2}

Factorise :- 64a^{3} – 27b^{3} – 144a^{2}b + 108ab^{2}

Factorise :-

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

Verify :- x^{3} + y^{3} = (x + y) (x^{2} – xy + y^{2})

Verify :- x^{3} – y^{3} = (x – y) (x^{2} + xy + y^{2})

Factorise :- 27y^{3} + 125z^{3}

Factorise :- 64m^{3} – 343n^{3}

Factorise :- 27x^{3} + y^{3} + z^{3} – 9xyz

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

If x + y + z = 0, show that x^{3} + y^{3} + z^{3} = 3xyz.

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

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

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

Area : 25a^{2} – 35a + 12 |

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

Area : 35y^{2} + 13y –12 |

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

Volume : 3x^{2} – 12x |

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

Volume : 12ky^{2} + 8ky – 20k |

## Chapter 2: Polynomials

#### NCERT Mathematics Class 9

#### Textbook solutions for 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.

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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.

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