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

Following expression is polynomial in one variable or not? State reason for your answer.

4x² - 3x + 7

Following expression is polynomial in one variable or not? State reason for your answer.

`y^2+sqrt2`

Following expression is polynomial in one variable or not? State reason for your answer.

`3sqrtt + tsqrt2`

Following expression is polynomial in one variable or not? State reason for your answer.

`y+2/y`

Following expression is polynomial in one variable or not? State reason for your answer.

x¹⁰ + y³ + t⁵⁰

Write the coefficient of x² in the following:-

`2+x^2+x`

Write the coefficient of x² in the following:-

2 - x² + x³

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 polynomial:-

`5t - sqrt7`

Write the degree of the following polynomial:-

3

Classify the following as linear, quadratic and cubic polynomial:

x² + x

Classify the following as linear, quadratic, and cubic polynomial:

x - x³

Classify the following as linear, quadratic and cubic polynomial:

y + y² + 4

Classify the following as linear, quadratic and cubic polynomial:

1 + x

Classify the following as linear, quadratic and cubic polynomial:

3t

Classify the following as linear, quadratic and cubic polynomial:

r²

Classify the following as linear, quadratic and cubic polynomial:

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 polynomial:-

p(y) = y² – y + 1

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

p(t) = 2 + t + 2t² – t³

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

p(x) = x³

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

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 are 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) = 3x² – 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.

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

x³ + x² + x + 1

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

x⁴ + x³ + x² + x + 1

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

x⁴ + 3x³ + 3x² + x + 1

Determine the following polynomial 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 identity to find the following product:

(x + 4) (x + 10) 

Use suitable identity to find the following product:

(x + 8) (x – 10)

Use suitable identity to find the following product:

(3x + 4) (3x – 5)

Use suitable identity to find the following product:

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

Use suitable identity to find the following product:

(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 identity:

9x² + 6xy + y² 

Factorise the following using appropriate identity:

4y² – 4y + 1

Factorise the following using appropriate identity:

`x^2 - y^2/100`

Expand the following, using suitable identity:

(x + 2y + 4z)² 

Expand the following, using suitable identity:

(2x – y + z)² 

Expand the following, using suitable identity:

(–2x + 3y + 2z)²

Expand the following, using suitable identity:

(3a – 7b – c)² 

Expand the following, using suitable identity:

(–2x + 5y – 3z)²

Expand the following, using suitable identity:

`[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 identity:

(99)³ 

Evaluate the following using suitable identity:

(102)³ 

Evaluate the following using suitable identity:

(998)³ 

Factorise the following:

8a³ + b³ + 12a²b + 6ab²

Factorise the following:

8a³ – b³ – 12a²b + 6ab²

Factorise the following:

27 – 125a³ – 135a + 225a²

Factorise the following:

64a³ – 27b³ – 144a²b + 108ab² 

Factorise the following:

`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 the following:

27y³ + 125z³

Factorise the following:

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 expression for the length and breadth of the following rectangle, in which their area are given:

Give possible expression for the length and breadth of the following rectangle, in which their area is given:

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

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