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

Which one of the following is a polynomial?

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

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

Degree of the zero polynomial is ______.

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

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

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

Zero of the zero polynomial is ______.

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

One of the zeroes of the polynomial 2x² + 7x – 4 is ______.

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

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

x + 1 is a factor of the polynomial ______.

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

The value of 249² – 248² is ______.

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

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

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

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

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

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

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

8

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

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

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

`1 - sqrt(5)x`

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

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

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

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

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

`1/(x + 1)`

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

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

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

`1/(2x)`

A binomial can have atmost two terms

Every polynomial is a binomial.

A binomial may have degree 5

Zero of a polynomial is always 0

A polynomial cannot have more than one zero

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

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

x² + x + 1

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

y³ – 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.

x² – 2xy + y² + 1

Determine the degree of the following polynomial:

2x – 1

Determine the degree of the following polynomial:

–10

Determine the degree of the following polynomial:

x³ – 9x + 3x⁵ 

Determine the degree of the following polynomial:

y³(1 – y⁴)

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 x³ 

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

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

Write the coefficient of x² in the following:

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

Write the coefficient of x² in the following:

3x – 5

Write the coefficient of x² in the following:

(x – 1)(3x – 4)

Write the coefficient of x² in the following:

(2x – 5)(2x² – 3x + 1)

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

2 – x² + x³ 

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

3x³ 

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 – 5y² 

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:

y³ – y

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

1 + x + x²

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

t² 

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 3x³ – 4x² + 7x – 5, when x = 3 and also when x = –3.

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

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

p(x) = 10x – 4x² – 3

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

p(y) = (y + 2)(y – 2)

–3 is a zero of x – 3

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

`(-4)/5` is a zero of 4 – 5y

0 and 2 are the zeroes of t² – 2t

–3 is a zero of y² + y – 6

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: x⁴ + 1; x – 1

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

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

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

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`

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

p(x) = x³ – 5x² + 4x – 3, g(x) = x – 2

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

p(x) = 2x³ – 11x² – 4x + 5, g(x) = 2x + 1

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

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

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

3x² + 6x – 24

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

4x² + x – 2

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

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

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

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

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

Factorise:

x² + 9x + 18

Factorise:

6x² + 7x – 3

Factorise:

2x² – 7x – 15

Factorise:

84 – 2r – 2r²

Factorise:

2x³ – 3x² – 17x + 30

Factorise:

x³ – 6x² + 11x – 6

Factorise:

x³ + x² – 4x – 4

Factorise:

3x³ – x² – 3x + 1

Using suitable identity, evaluate the following:

103³

Using suitable identity, evaluate the following:

101 × 102

Using suitable identity, evaluate the following:

999²

Factorise the following: 

4x² + 20x + 25

Factorise the following:

9y² – 66yz + 121z²

Factorise the following:

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

Factorise the following:

9x² – 12x + 3

Factorise the following:

9x² – 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:

9x² + 4y² + 16z² + 12xy – 16yz – 24xz

Factorise the following:

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

Factorise the following:

16x² + 4y² + 9z² – 16xy – 12yz + 24xz

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

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 – 64a³ – 12a + 48a²

Factorise the following:

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

Find the following product:

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

Find the following product:

(x² – 1)(x⁴ + x² + 1)

Factorise:

1 + 64x³ 

Factorise:

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

Find the following product:

(2x – y + 3z)(4x² + y² + 9z² + 2xy + 3yz – 6xz)

Factorise:

a³ – 8b³ – 64c³ – 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 x³ + y³ – 12xy + 64, when x + y = – 4

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

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

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

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.

If both x – 2 and `x - 1/2` are factors of px² + 5x + r, show that p = r.

Without actual division, prove that 2x⁴ – 5x³ + 2x² – x + 2 is divisible by x² – 3x + 2. [Hint: Factorise x² – 3x + 2]

Simplify (2x – 5y)³ – (2x + 5y)³.

Multiply x² + 4y² + z² + 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 a³ + b³ + c³ – 3abc = – 25.

Prove that (a + b + c)³ – a³ – b³ – c³ = 3(a + b)(b + c)(c + a).