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

Exercise 2.1

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

1. \[4{{x}^{2}}-3x+7\]

2. \[{{y}^{2}}+\sqrt{2}\]

3. \[3\sqrt{t}+t\sqrt{2}\]

4. \[y+\frac{2}{y}\]

5. \[y+2{{y}^{-1}}\]

Ans:

A polynomial in one variable refers to an expression where the exponent of the variable is a whole number.

1. \[4{{x}^{2}}-3x+7\]

In this polynomial, only one variable is involved which is ‘$x$ ’ and the exponents of the variable are all whole numbers.

Therefore, the given expression is a polynomial in one variable ‘$x$ ’.

2. \[{{y}^{2}}+\sqrt{2}\]

In this polynomial, only one variable is involved which is ‘$y$ ’ and the exponent of the variable is a whole number.

Therefore, the given expression is a polynomial in one variable ‘$y$ ’.

3. \[3\sqrt{t}+t\sqrt{2}\]

In this expression, it is given that the exponent of variable $t$ in term ‘$3\sqrt{t}$ ’ is $\frac{1}{2}$. This exponent is not a whole number. Therefore, the given algebraic expression is not a polynomial in one variable.

4. \[y+\frac{2}{y}\]

We can rewrite this expression as: $y+2{{y}^{-1}}$.

In this expression, it is given that the exponent of the variable $y$ in term ‘$2{{y}^{-1}}$ ’ is $-1$. This exponent is not a whole number. Therefore, the given algebraic expression is not a polynomial in one variable.

5. ${{x}^{10}}+{{y}^{3}}+{{t}^{50}}$ 

In this polynomial, there are $3$ variables involved which are‘$x,y,t$’.

Therefore, the given algebraic expression is not a polynomial in one variable.

2. Write the coefficients of ${{x}^{2}}$in each of the following:

1. $2+{{x}^{2}}+x$ 

2. $2-{{x}^{2}}+{{x}^{3}}$ 

3. $\frac{\pi }{2}{{x}^{2}}+x$ 

4. $\sqrt{2}x-1$ 

Ans: A coefficient is an integer that is multiplied by the variable of a only one term or the terms of a polynomial.

1. $2+{{x}^{2}}+x$

We can rewrite this expression as: $2+1({{x}^{2}})+x$.

Hence, the coefficient of ${{x}^{2}}$ is $1$.

2. $2-{{x}^{2}}+{{x}^{3}}$

We can rewrite this expression as: $2-1({{x}^{2}})+{{x}^{3}}$.

Hence, the coefficient of ${{x}^{2}}$ is $-1$.

3. $\frac{\pi }{2}{{x}^{2}}+x$

In the given expression, the coefficient of ${{x}^{2}}$ is $\frac{\pi }{2}$ .

4. $\sqrt{2}x-1=0{{x}^{2}}+\sqrt{2}x-1$ 

In the given expression, the coefficient of ${{x}^{2}}$ is $0$.

3. Give one example each of a binomial of degree \[35\], and of a monomial of degree $100$ .

Ans: A binomial of degree $35$ refers to a polynomial with two terms and one of the terms has a highest degree of $35$.

Example: ${{x}^{35}}+{{x}^{34}}$

A monomial of degree $100$ refers to a polynomial with only one term and it has a highest degree of $100$.

Example: ${{x}^{100}}$ 

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

(I) $5{{x}^{3}}+4{{x}^{2}}+7x$ 

(II) $4-{{y}^{2}}$ 

(III) $5t-\sqrt{7}$ 

(IV) $3$ 

Ans: The degree of a polynomial refers to the highest power of a variable in the polynomial.

(i) $5{{x}^{3}}+4{{x}^{2}}+7x$

Here, the highest power of the given variable ‘$x$ ’ is $3$. Hence, the degree of this polynomial is  $3$ .

(ii) $4-{{y}^{2}}$

Here, the highest power of the given variable ‘$y$ ’ is  $2$ . Hence, the degree of this polynomial is  $2$ .

(iii) $5t-\sqrt{7}$

Here, the highest power of the given variable ‘$t$ ’ is  $1$ . Hence, the degree of this polynomial is  $1$ .

(iv) $3$

Here, $3$is a constant polynomial. We know the degree of a constant polynomial is always  $0$ . Hence, the degree of this polynomial is  $0$ .

5. Classify the following as linear, quadratic and cubic polynomial:

${{x}^{2}}+x$ 

(i)  $x-{{x}^{3}}$ 

(ii) $y+{{y}^{2}}+4$ 

(iii) $1+x$

(iv) $3t$ 

(v) ${{r}^{2}}$ 

(iv) $7{{x}^{3}}$ 

Ans: The highest exponential power of the variable in a polynomial equation is known as the degree of a polynomial. 

A linear polynomial is a polynomial whose degree is ‘$1$ ’.

A quadratic polynomial is a polynomial whose degree is ‘$2$ ’.

A cubic polynomial is a polynomial whose degree is ‘$3$ ’.

(i) ${{x}^{2}}+x$  

The given expression has a variable $x$ and its degree is $2$.

Hence, it is a quadratic polynomial.

(ii) $x-{{x}^{3}}$   

The given expression has a variable $x$ and its degree is $3$.

Hence, it is a cubic polynomial.

(iii) $y+{{y}^{2}}+4$

The given expression has a variable $y$ and its degree is $2$.

Hence, it is a quadratic polynomial.

(iv) $1+x$ $y+{{y}^{2}}+4$

The given expression has a variable $x$ and its degree is $1$.

Hence, it is a linear polynomial.

(v) $3t$ 

The given expression has a variable $t$ and its degree is $1$.

Hence, it is a linear polynomial.

(vi) ${{r}^{2}}$ 

The given expression has a variable $r$ and its degree is $2$.

Hence, it is a quadratic polynomial.

(vii) $7{{x}^{3}}$

The given expression has a variable $x$ and its degree is $3$.

Hence, it is a cubic polynomial.

What do you learn from the Class 9 Maths Chapter 2 Exercise 2.1?

The NCERT textbook for Class 9 Maths, Chapter 2 introduces you to polynomials. In section 2.1, you will get to learn about polynomials, how an algebraic expression is structured, how you can find out if a variable is polynomial from its exponential powers as presented in an equation, etc. The Class 9 Maths Chapter 2 exercise 2.1, introduces you to the most common pattern of questions posed on polynomials, in the CBSE exams.

Polynomials form an important part of the CBSE Class 9 Maths syllabus. If you consult the last few years question papers as well, you shall notice that polynomials of Class 9 exercise 2.1 form a decent percentage of the marking scope of the Maths paper. 

NCERT Class 9 Maths Chapter 2 Exercise 2.1

The CBSE Class 9 Maths syllabus finally introduces students to the basics of high school Mathematics. If you are one of those students then you should know that you may very well be using some of the formulae you learn, well into your future career. Exercise 2.1 class 9 presents a tricky set of problems that are commonly present in the CBSE Class 9 Maths final paper. Students who aspire to score top marks in their CBSE Class 9 Maths paper, must possess masterful knowledge of the subject by their exam time. Scoring beyond expectations in a subject as vast and complex as Class 9 Maths is no easy task. 

When it comes to practicing regularly, it pays to know all the accurate steps for deducing the mathematical solution. In this regard, Vedantu’s free PDF download for the solutions to the NCERT Class 9 Maths, ex 2.1 class 9, can help you to greatly refine your polynomial problem-solving skills. The PDF solutions are portable in nature and can be carried with you on your smartphone, laptop, etc. and referred at convenience for cross-checking your problem-solving and skill improvement. 

Download Vedantu NCERT Book Solution to get a better understanding of all the exercises questions.

Why avail the Vedantu NCERT Exercise 2.1 class 9 maths solutions free PDF?

The NCERT Solutions for Class 9 Maths Chapter 2 polynomials exercise 2.1, are broadly discussed in the PDF, with well-defined steps. Vedantu’s in-house tutors and subject experts have presented the solutions in an easy-to-understand, CBSE answer format. Preparing for your Maths exam has never been easier with Vedantu by your side. Class 9 exercise 2.1 paved the way for algebraic identities, remainder theorem, polynomial factorization, etc; and, now you can master them all with Vedantu.

NCERT Solutions for Class 9 Maths

• Chapter 1 - Number Systems

• Chapter 2 - Polynomials

• Chapter 3 - Coordinate Geometry

• Chapter 4 - Linear Equations in Two Variables

• Chapter 5 - Introduction to Euclids Geometry

• Chapter 6 - Lines and Angles

• Chapter 7 - Triangles

• Chapter 8 - Quadrilaterals

• Chapter 9 - Areas of Parallelograms and Triangles

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Heron's Formula

• Chapter 13 - Surface Areas and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Access Other Exercise Solutions of Class 9 Maths Chapter 2 Polynomials