Class 9 NCERT Solutions – Chapter 2 Polynomials – Exercise 2.1

Question 1. 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) The algebraic expression 4x² – 3x + 7 can be written as 4x² – 3x + 7x⁰

As we can see, all exponents of x are whole numbers, 

So, the given expression 4x² – 3x + 7 is polynomial in one variable.

(ii) The algebraic expression y² + √2 can be written as y² + √2y⁰

As we can see, all exponents of y are whole numbers, 

So, the given expression y² + √2 is polynomial in one variable.

(iii) The algebraic expression 3 √t + t√2 can be written as 3 t^1/2 + √2.t

As we can see, one exponent of t is 1/2, which is not a whole number,

So, the given expression 3 √t + t√2 is not a polynomial in one variable.

(iv) The algebraic expression y + 2/y can be written as y + 2.y^-1

As we can see, one exponent of y is -1, which is not a whole number,

So, the given expression y+ 2/y is not a polynomial in one variable.

(v) The given algebraic expression is x¹⁰+ y³+ t⁵⁰

As we can see, the expression contains three variables i.e x, y, and t,

So, the given expression x¹⁰ + y³ + t⁵⁰ 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/2 x² + x
(iv) √2x – 1

Solution: 

(i) The given algebraic expression is 2 + x² + x

As we can clearly see, the coefficient of x² is 1.

(ii) The given algebraic expression is 2 – x² + x³

As we can clearly see, the coefficient of x² is -1.

(iii) The given algebraic expression is pi/2 x² + x

As we can clearly see, the coefficient of x² is pi/2.

(iv) The given algebraic expression is √2 x — 1

As we can clearly see, the coefficient of x² is 0.

Question 3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution: 

A Binomial having degree 35 is 4x³⁵ + 50

A Monomial having degree 100 is 3t¹⁰⁰pi

Question 4. Write the degree of each of the following polynomials

(i) 5x³ + 4x² + 7x
(ii) 4 – y²
(iii) 5t – √7
(iv) 3

Solution: 

The highest power of a variable in the given expression is known as the Degree of the polynomial 

(i) The given expression is 5x³ + 4x² + 7x

As we can clearly see, the highest power of variable x is 3,

So, the degree of given polynomial 5x³+4x² + 7x is 3.

(ii) The given expression is 4 – y²

As we can clearly see, the highest power of variable y is 2,

So, the degree of given polynomial 4 – y² is 2.

(iii) The given expression is 5t – √7

As we can clearly see, the highest power of variable t is 1,

So, the degree of given polynomial 5t – √7 is 1.

(iv) The given expression 3 can be written as 3x⁰

As we can clearly see, the highest power of variable x is 0,

So, the degree of given polynomial 3 is 0.

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: 

(i) Since the degree of given polynomial x² + x is 2,

So, it is a Quadratic Polynomial.

(ii) Since the degree of given polynomial x – x³ is 3,

So, it is a Cubic Polynomial.

(iii) Since the degree of given polynomial y + y² + 4 is 2,

So, it is a Quadratic Polynomial.

(iv) Since the degree of given polynomial 1 + x is 1,

So, it is a Linear Polynomial.

(v) Since the degree of given polynomial 3t is 1, 

So, it is a Linear Polynomial.

(vi) Since the degree of given polynomial r² is 2,

So, it is a Quadratic Polynomial.

(vii) Since the degree of given polynomial 7x³ is 3,

So, it is a Cubic Polynomial.