#### Chapters

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equation In Two Variables

Chapter 5: Introduction To Euclid's Geometry

Chapter 6: Lines & Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Areas of Parallelograms & Triangles

Chapter 10: Circles

Chapter 11: Construction

Chapter 12: Heron's Formula

Chapter 13: Surface Area & Volumes

Chapter 14: Statistics & Probability

## Chapter 2: Polynomials

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

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

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)`

`sqrt(2)` is a polynomial of degree ______.

2

0

1

`1/2`

Degree of the polynomial 4x^{4} + 0x^{3} + 0x^{5} + 5x + 7 is ______.

4

5

3

7

Degree of the zero polynomial is ______.

0

1

Any natural number

Not defined

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

0

1

`4sqrt(2)`

`8sqrt(2) + 1`

The value of the polynomial 5x – 4x^{2} + 3, when x = –1 is ______.

– 6

6

2

–2

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

3

2x

0

6

Zero of the zero polynomial is ______.

0

1

Any real number

Not defined

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

`-2/5`

`-5/2`

`2/5`

`5/2`

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

2

`1/2`

`-1/2`

`-2`

If x^{51} + 51 is divided by x + 1, the remainder is ______.

0

1

49

50

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

–3

4

2

–2

x + 1 is a factor of the polynomial ______.

x

^{3}+ x^{2}– x + 1x

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

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

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

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

5 + x

5 – x

5x – 1

10x

The value of 249^{2} – 248^{2} is ______.

1

^{2}477

487

497

The factorisation of 4x^{2} + 8x + 3 is ______.

(x + 1)(x + 3)

(2x + 1)(2x + 3)

(2x + 2)(2x + 5)

(2x –1)(2x –3)

Which of the following is a factor of (x + y)^{3} – (x^{3} + y^{3})?

x

^{2}+ y^{2}+ 2xyx

^{2}+ y^{2}– xyxy

^{2}3xy

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

1

9

18

27

If `x/y + y/x = -1 (x, y ≠ 0)`, the value of x^{3} – y^{3} is ______.

1

–1

0

`1/2`

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

0

`1/sqrt(2)`

`1/4`

`1/2`

If a + b + c = 0, then a^{3} + b^{3} + c^{3} is equal to ______.

0

abc

3abc

2abc

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

Which of the following expression are polynomials? Justify your answer:

8

Which of the following expression are polynomials? Justify your answer:

`sqrt(3)x^2 - 2x`

Which of the following expression are polynomials? Justify your answer:

`1 - sqrt(5)x`

Which of the following expression are polynomials? Justify your answer:

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

Which of the following expression are polynomials? Justify your answer:

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

Which of the following expression are polynomials? Justify your answer:

`1/(x + 1)`

Which of the following expression are polynomials? Justify your answer:

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

Which of the following expression are polynomials? Justify your answer:

`1/(x + 1)`

Which of the following expression are polynomials? Justify your answer:

`1/(x + 1)`

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

A binomial can have atmost two terms

True

False

Every polynomial is a Binomial.

True

False

A binomial may have degree 5

True

False

Zero of a polynomial is always 0

True

False

A polynomial cannot have more than one zero

True

False

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

True

False

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

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

x^{2} + x + 1

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

y^{3 }– 5y

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

xy + yz + zx

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

x^{2} – 2xy + y^{2} + 1

Determine the degree of the following polynomials:

2x – 1

Determine the degree of the following polynomials:

–10

Determine the degree of the following polynomials:

x^{3} – 9x + 3x^{5}

Determine the degree of the following polynomials:

y^{3}(1 – y^{4})

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

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

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

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

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

`pi/6 x + x^2 - 1`

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

3x – 5

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

(x – 1)(3x – 4)

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

(2x – 5)(2x^{2} – 3x + 1)

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

2 – x^{2} + x^{3}

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

3x^{3}

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

`5t - sqrt(7)`

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

4 – 5y^{2}

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

3

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

2 + x

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

y^{3} – y

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

1 + x + x^{2}

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

t^{2}

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

`sqrt(2)x - 1`

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

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

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

Find the value of the polynomial 3x^{3} – 4x^{2} + 7x – 5, when x = 3 and also when x = –3.

If p(x) = x^{2} – 4x + 3, then evaluate p(2) – p(–1) + `p(1/2)`.

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

p(x) = 10x – 4x^{2} – 3

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:

–3 is a zero of x – 3

True

False

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

True

False

`(-4)/5` is a zero of 4 – 5y

True

False

0 and 2 are the zeroes of t^{2} – 2t

True

False

–3 is a zero of y^{2} + y – 6

True

False

Find the zeroes of the polynomial in the following:

p(x) = x – 4

Find the zeroes of the polynomial in the following:

g(x) = 3 – 6x

Find the zeroes of the polynomial in the following:

q(x) = 2x – 7

Find the zeroes of the polynomial in the following:

h(y) = 2y

Find the zeroes of the polynomial:

p(x) = (x – 2)^{2} – (x + 2)^{2}

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: x^{4} + 1; x – 1

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where p(x) = x^{3} – 2x^{2} – 4x – 1, g(x) = x + 1

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where p(x) = x^{3} – 3x^{2} + 4x + 50, g(x) = x – 3

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

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

Check whether p(x) is a multiple of g(x) or not:

p(x) = x^{3} – 5x^{2} + 4x – 3, g(x) = x – 2

Check whether p(x) is a multiple of g(x) or not:

p(x) = 2x^{3} – 11x^{2} – 4x + 5, g(x) = 2x + 1

Show that: x + 3 is a factor of 69 + 11x – x^{2} + x^{3}.

Show that: 2x – 3 is a factor of x + 2x^{3} – 9x^{2} + 12.

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

3x^{2} + 6x – 24

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

4x^{2} + x – 2

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

4x^{2} + x – 2

Show that p – 1 is a factor of p^{10 }– 1 and also of p^{11} – 1.

For what value of m is x^{3} – 2mx^{2} + 16 divisible by x + 2?

If x + 2a is a factor of x^{5} – 4a^{2}x^{3} + 2x + 2a + 3, find a.

Find the value of m so that 2x – 1 be a factor of 8x^{4} + 4x^{3} – 16x^{2} + 10x + m.

If x + 1 is a factor of ax^{3} + x^{2} – 2x + 4a – 9, find the value of a.

Factorise: x^{2} + 9x + 18

Factorise: 6x^{2} + 7x – 3

Factorise: 2x^{2} – 7x – 15

Factorise: 84 – 2r – 2r^{2}

Factorise: 2x^{3} – 3x^{2} – 17x + 30

Factorise: x^{3} – 6x^{2} + 11x – 6

Factorise: x^{3} + x^{2} – 4x – 4

Factorise: 3x^{3} – x^{2} – 3x + 1

Using suitable identity, evaluate the following:

103^{3}

Using suitable identity, evaluate the following:

101 × 102

Using suitable identity, evaluate the following:

999^{2}

Factorise the following:

`4x^2 + 20x + 25`

Factorise the following:

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

Factorise the following:

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

Factorise the following :

9x^{2} – 12x + 3

Factorise the following:

9x^{2} – 12x + 4

Expand the following:

`(4a - b + 2c)^2`

Expand the following:

`(3a - 5b - c)^2`

Expand the following:

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

Factorise the following:

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

Factorise the following:

25x^{2} + 16y^{2} + 4z^{2} – 40xy + 16yz – 20xz

Factorise the following:

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

If a + b + c = 9 and ab + bc + ca = 26, find a^{2} + b^{2} + c^{2}.

Expand the following:

(3a – 2b)^{3}

Expand the following:

`(1/x + y/3)^3`

Expand the following:

`(4 - 1/(3x))^3`

Factorise the following:

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

Factorise the following:

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

Find the following products:

`(x/2 + 2y)(x^2/4 - xy 4y^2)`

Find the following products:

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

Factorise: 1 + 64x^{3}

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

Find the following product:

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

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

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

Without actually calculating the cubes, find the value of:

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

Without actually calculating the cubes, find the value of:

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

Without finding the cubes, factorise:

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

Find the value of x^{3} + y^{3} – 12xy + 64,when x + y = – 4.

Find the value of x^{3} – 8y^{3} – 36xy – 216, when x = 2y + 6

Give possible expressions for the length and breadth of the rectangle whose area is given by 4a^{2} + 4a – 3.

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

If the polynomials az^{3} + 4z^{2} + 3z – 4 and z^{3} – 4z + o leave the same remainder when divided by z – 3, find the value of a.

The polynomial p(x) = x^{4} – 2x^{3} + 3x^{2} – 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.

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

Without actual division, prove that 2x^{4} – 5x^{3} + 2x^{2} – x + 2 is divisible by x^{2} – 3x + 2.

Simplify: (2x – 5y)^{3} – (2x + 5y)^{3}.

Multiply x^{2} + 4y^{2} + z^{2} + 2xy + xz – 2yz by (– z + x – 2y).

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.

If a + b + c = 5 and ab + bc + ca = 10, then prove that a^{3} + b^{3} + c^{3} – 3abc = – 25.

Prove that (a + b + c)^{3} – a^{3} – b^{3 }– c^{3} = 3(a + b)(b + c)(c + a).

## Chapter 2: Polynomials

## 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