NCERT solutions for Class 9 Maths chapter 2 - Polynomials [Latest edition]

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 x² in the following:-

`2+x^2+x`

Write the coefficient of x² in the following:-

`2-x^2+x^3`

Write the coefficient of x² in the following:-

`pi/2x^2+x`

Write the coefficient of x² 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:-

5x³ + 4x² +7x

Write the degree of the following polynomial:-

4 - y²

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) x² + x

(ii) x – x³

(iii) y + y² + 4

(iv) 1 + x

(v) 3t

(vi) r²

(vii) 7x³

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

x = 0

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

x = –1

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

 x = 2

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

p(y) = y² – y + 1

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

p(t) = 2 + t + 2t² – t³

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

p(x) = x³

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) = x² – 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) = x², 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.

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

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

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

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

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

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

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

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

x³ + x² + x + 1

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

x⁴ + x³ + x² + x + 1

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

x⁴ + 3x³ + 3x² + 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) = 2x³ + x² – 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) = x³ + 3x² + 3x + 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) = x³ − 4x² + 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) = x² + 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) = kx² – 3x + k

Factorise :- 12x² – 7x + 1

Factorise :- 2x² + 7x + 3

Factorise :- 6x² + 5x – 6

Factorise :- 3x² – x – 4

Factorise :- x³ – 2x² – x + 2

Factorise :- x³ – 3x² – 9x – 5

Factorise :- x³ + 13x² + 32x + 20

Factorise :- 2y³ + y² – 2y – 1

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:- 9x² + 6xy + y² 

Factorise the following using appropriate identities:- 4y² – 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:- 4x² + 9y² + 16z² + 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 :- 8a³ + b³ + 12a²b + 6ab² 

Factorise :- 8a³ – b³ – 12a²b + 6ab²

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

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

Factorise :-

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

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

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

Factorise :- 27y³ + 125z³

Factorise :- 64m³ – 343n³ 

Factorise :- 27x³ + y³ + z³ – 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 x³ + y³ + z³ = 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:-

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

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

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