Class 9 NCERT Solutions Chapter 2 Polynomials Exercise 2.1 (original) (raw)

Last Updated : 23 Jul, 2025

Chapter 2 of the Class 9 NCERT Mathematics textbook, titled "Polynomials," introduces the concept of polynomials, including their types, degrees, and operations. This chapter explores the properties of polynomials and their applications in algebraic expressions. Exercise 2.1 focuses on fundamental problems involving polynomials, such as identifying and evaluating polynomial expressions, and understanding their basic properties.

NCERT Solutions for Class 9 - Chapter 2 Polynomials - Exercise 2.1

This section provides detailed solutions for Exercise 2.1 from Chapter 2 of the Class 9 NCERT Mathematics textbook. The exercise includes problems that involve evaluating polynomials for given values, performing polynomial operations, and identifying polynomial characteristics. Solutions are provided step-by-step to help students understand and apply the concepts of polynomials effectively.

**Question 1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

****(i) 4x** 2 - 3x + 7
****(ii) y** 2 + √2
****(iii) 3√t + t√2**
****(iv) y + 2/y**
****(v) x** 10 **+ y 3 **+ t 50

**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 4x 2 - 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 y 2 + √2 is polynomial in one variable.

****(iii)** The algebraic expression 3 √t + t√2 can be written as 3 t1/2 + √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 x 10 **+ y 3 **+ t 50 **is not a polynomial in one variable.

**Question 2. Write the coefficients of x 2 in each of the following

****(i) 2 + x** 2 + x
****(ii) 2 - x** 2 + x 3
****(iii) pi/2 x** 2 + x
****(iv) √2x - 1**

**Solution:

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

As we can clearly see, **the coefficient of x 2 **is 1.

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

As we can clearly see, **the coefficient of x 2 **is -1.

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

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

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

As we can clearly see, **the coefficient of x 2 **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 4x 35 + 50

A Monomial having degree 100 is **3t 100 pi

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

****(i) 5x** 3 **+ 4x 2 + 7x
****(ii) 4 - y** 2
****(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 5x 3 +4x 2 + 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 - y 2 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) x** 2 **+ x
****(ii) x - x** 3
****(iii) y + y** 2 **+ 4
****(iv) 1 + x**
****(v) 3t**
****(vi) r** 2
****(vii) 7x** 3

**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.

Summary

Chapter 2 of the Class 9 NCERT Mathematics textbook, "Polynomials," introduces the fundamentals of polynomials, including their types, degrees, and operations. Exercise 2.1 provides practice problems on evaluating polynomials, performing basic operations like addition and subtraction, and understanding polynomial properties. The exercise helps students solidify their understanding of polynomial expressions and their applications.

What is a polynomial?

A polynomial is an algebraic expression consisting of one or more terms, each of which includes a variable raised to a non-negative integer power and a coefficient. For example, 3x2 - 4x + 5 is a polynomial.

How do you determine the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 4x3 - 2x2 +7, the degree is 3.

What is the difference between a polynomial and a non-polynomial expression?

A polynomial expression consists of terms with non-negative integer exponents of variables and real number coefficients, whereas non-polynomial expressions may include negative or fractional exponents, or variables in the denominator.