Exercise 2.1 

Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x² – 3x + 7 

(ii) y² + √2

(iii)3√t + t√2

(iv) y +2/y

(v) x¹⁰ + y³ + t⁵⁰

Solution:

(i) 4x² – 3x + 7 

It is polynomial in one variable. Because the power of variable is natural number. 

(ii) y² + √2 

It is polynomial in one variable. Because the power of variable is natural number. 

(iii)3√t + t√2 

It is not a polynomial in one variable. Because the power of variable is not a natural number, it is a fractional number. 

(iv) y +2/y

It is not a polynomial in one variable.

(v) x¹⁰ + y³ + t^​50

It is not a polynomial in one variable. while It is a polynomial in three variable. 

Q2. Write the coefficients of x² in each of the following:

(i) 2 + x² + x 

(ii) 2 – x² + x3 (

iii) π/2 x² + x

(iv) √2x −1

Solution: 

(i) 2 + x² + x 

Coefficient of x² = 1 

(ii) 2 – x² + x3 

Coefficient of x² = –1 ​

(iii) π/2 x² + x

Coefficient of x² = π/2

(iv) √2x −1

Coefficients of x² = 0 [Because There is no x² So coefficient will be 0]

Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution: 

A binomial of degree 35 

⇒ 2x³⁵ + 5y ​

Note: Binomial means the expression of two terms. Like x + 5, 3a - 2b, 3t + 7 etc. 

A monomial of degree 100

⇒ 3y¹⁰⁰

Note: Monomial means the expression of single term, like 3x, 5t, y, 3xy etc. 

Q4. Write the degree of each of the following polynomials:
(i) 5x³ + 4x² + 7x

(ii) 4 – y²

(iii) 5t – √7

(iv) 3

Solution: 

(i) 5x³ + 4x² + 7x 

Ans: The degree of polynomial = 3               [Highest power]

(ii) 4 – y^​2

Ans: The degree of polynomial = 2

(iii) 5t – √7 

Ans: The degree of polynomial = 1

(iv) 3

Ans: The degree of polynomial = 0

[Note: There is no variable therefore degree = 0] 

Q5. Classify the following as linear, quadratic and cubic polynomials:

(i) x^​2 + x 

(ii) x – x³ 

(iii) y + y² + 4

(iv) 1 + x

(v) 3t

(vi) r² 

(vii) 7x²

Solution:

(i) x^​2 + x 

Ans: Quadratic polynomial

(ii) x – x³ 

Ans: Cubic polynomial

(iii) y + y² + 4 

Ans: Quadratic polynomial

(iv) 1 + x

Ans : Linear polynomial

(v) 3t 

Ans : Linear polynomial

(vi) r² 

Ans: Quadratic polynomial

(vii) 7x²

Ans: Cubic polynomial