Solution of Chapter 2. Polynomials (NCERT Mathematics Exemplar Book)

Which of the following expressions are polynomials? Justify your answer.

8

Which of the following expressions are polynomials? Justify your answer.

√3x² - 2x

Which of the following expressions are polynomials? Justify your answer.

1 - √5x

Which of the following expressions are polynomials? Justify your answer.

Which of the following expressions are polynomials? Justify your answer.

Which of the following expressions are polynomials? Justify your answer.

Which of the following expressions are polynomials? Justify your answer.

Which of the following expressions are polynomials? Justify your answer.

Write whether the following statements are True or False. Justify your answer.

A binomial can have at most two terms

Write whether the following statements are True or False. Justify your answer.

Every polynomial is a binomial.

Write whether the following statements are True or False. Justify your answer.

A binomial may have degree 5

Write whether the following statements are True or False. Justify your answer.

Zero of a polynomial is always 0.

Write whether the following statements are True or False. Justify your answer.

A polynomial cannot have more than one zero

Write whether the following statements are True or False. Justify your answer.

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

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

(i) x² + x + 1

(ii) y³ – 5y

(iii) xy + yz + zx

(iv) x² – 2xy + y² + 1

Determine the degree of each of the following polynomials:

(i) 2x – 1

(ii) –10

(iii) x³ – 9x + 3x⁵

(iv) y³ (1 – y⁴)

Write the coefficient of x² in each of the following:

(i)

(ii) 3x – 5

(iii) (x –1) (3x – 4)

(iv) (2x – 5) (2x² – 3x + 1)

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

(i) 2 – x² + x³ (ii) 3x³

(iii) 5t – √7 (iv) 4 – 5y²

(v) 3 (vi) 2 + x

(vii) y³ – y (viii) 1 + x + x²

(ix) t² (x) √2x – 1

Give an example of a polynomial, which is:

(i) monomial of degree 1

(ii) binomial of degree 20

(iii) trinomial of degree 2

Verify whether the following are true or false:

(i) –3 is a zero of x – 3

(ii) is a zero of 3x + 1

(iii) is a zero of 4 –5y

(iv) 0 and 2 are the zeroes of t² – 2t

(v) –3 is a zero of y² + y – 6

Find the zeroes of the polynomial in each of the following:

(i) p(x) = x – 4

(ii) g(x) = 3 – 6x

(iii) q(x) = 2x –7

(iv) h(y) = 2y

Find the zeroes of the polynomial:

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

Show that:

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

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

Factorise:

x² + 9x + 18

Factorise:

6x² + 7x – 3

Factorise:

2x² – 7x – 15

Factorise:

84 – 2r – 2r²

Factorise:

Factorise:

Factorise:

Factorise:

Using suitable identity, evaluate the following:

(i) 103³

(ii) 101 × 102

(iii) 999²

Factorise the following:

Factorise the following:

Expand the following:

Factorise the following:

Expand the following:

Factorize the following:

Find the following products:

Factorise:

Find the following product:

Factorise:

Without actually calculating the cubes, find the value of:

(i).

(ii). (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