Polynomials ⇒⇒ Exercise 2.1

Question 1 (i)

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x² - 3x +7

Solution :

True .

Because , Given polynomial 4x² - 3x +7 has only one variable which is 'x'

The exponents of x are 2, 1, and 0 (since 7 = 7$x^0 $), which are all non-negative integers. So this is a polynomial in one variable .

Question 1 (ii)

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(ii) y² + $ { \sqrt 2} $

Solution :

True .

Because , Given polynomial y² + $ { \sqrt 2} $ has only one variable which is 'y'

The exponents of y are 2 and 0 (since $ { \sqrt 2} $ = $ { \sqrt 2}y^0 $), which are all non-negative integers. So this is a polynomial in one variable .

Question 1 (iii)

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(iii) 3$ { \sqrt t} + t { \sqrt 2} $

Solution :

No .

Because , Given expression 3$ {\sqrt t}+ t { \sqrt 2} $ has only one variable which is 't' but the exponent of variable 3$ {\sqrt t}$ is ${ 1\over 2} $ which is not a non-negative integer. Therefore this expression is not a polynomial.

Question 1 (iv)

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(iv) p(y) = y + $ { 2 \over y} $

Solution :

No .

Because , Given expression y + $ { 2 \over y} $ has only one variable which is 'y' but the exponent of variable $ { 2 \over y} $ is -1 which is not a whole number. Therefore this expression is not a polynomial.

Question 1 (v)

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(v) x¹⁰ + y³ + t⁵⁰

Solution :

No .

Because , Given expression x¹⁰ + y³ + t⁵⁰ has three variables, i.e. x, y and t, so it is not a polynomial in one variable.

Question 2

Write the coefficients of x² in each of the following:
(i) 2 + x² + x
(ii) 2 - x² + x³
(iii) $ { \pi \over 2} $ x² + x
(iv) $ { \sqrt 2} $x - 1

Solution :

(i) Answer :

Given, 2 + x² + x

Here coefficients of x² = 1

(ii) Answer :

Given, 2 - x² + x³

Here coefficients of x² = -1

(iii) Answer :

Given, $ { \pi \over 2} $ x² + x

Here coefficients of x² = $ { \pi \over 2} $

(iv) Answer :

Given, $ { \sqrt 2} $x - 1

Here coefficients of x² = 0

Question 3

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution :

Binomial is a polynomial with two terms.And the degree of a polynomial is the highest power of the polynomial.

Example : 7x³⁵ + 12

And, In monomial, there is only one term in it.

Example : 7z¹⁰⁰

Question 4

Write the degree of each of the following polynomials:
(i) 5x³ + 4x² +7x
(ii) 4 - y²
(iii) 5t - ${ \sqrt 7} $
(iv) 3

Solution :

The degree of a polynomial is the highest power of the polynomial.

(i) Answer :

Given, 5x³ + 4x² +7x

Here highest power is 3, Hence, degree of a polynomial is 3

(ii) Answer :

Given, 4 - y²

Here highest power is 2, Hence,degree of a polynomial is 2

(iii) Answer :

Given, 5t - ${ \sqrt 7}$

Here highest power is 1, Hence, degree of a polynomial is 1

(iv) Answer :

Given, 3

There is no variable in the given polynomial and only a constant value 3 is there. Therefore, the degree of the polynomial is 0.
Degree of constant term is always zero.

Question 5

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

Solution :

Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively

(i) Answer :

Given, x² + x

Here highest power is 2, Hence, degree of a polynomial is 2. Therefore, it is a quadratic polynomial.

(ii) Answer :

Given, x - x³

Here highest power is 3, Hence, degree of a polynomial is 3 . Therefore, it is a cubic polynomial.

(iii) Answer :

Given, y + y² + 4

Here highest power is 2, Hence, degree of a polynomial is 2. Therefore, it is a quadratic polynomial.

(iv) Answer :

Given, 1 + x

Here highest power is 1, Hence, degree of a polynomial is 1. Therefore, it is a linear polynomial.

(v) Answer :

Given, 3t

Here highest power is 1, Hence, degree of a polynomial is 1. Therefore, it is a linear polynomial.

(vi) Answer :

Given, r²

Here highest power is 2, Hence, degree of a polynomial is 2. Therefore, it is a quadratic polynomial.

(vii) Answer :

Given, 7x³

Here highest power is 3, Hence, degree of a polynomial is 3. Therefore, it is a cubic polynomial.