 # NCERT Solutions for Class 10 Maths Chapter 2- Polynomials

## NCERT Solutions For Class 10 Maths Chapter 2 PDF Free Download

NCERT solutions Class 10 Maths Chapter 2 – Polynomials can be studied here. The subject experts of Maths have prepared these solutions to help students prepare well for their exams. They solve these solutions in such a way that it becomes easier for students to practice the questions of Chapter 2, Polynomials using NCERT Solutions. They make it simple to learn for students by adding stepwise solutions to these Maths NCERT Class 10 solutions.

NCERT Solutions for Class 10 Maths are an extremely important study resource for students. Solving these Polynomials NCERT solutions of class 10 Maths would help the students fetch good marks in board exams. Also, utmost importance to following the NCERT guidelines is focused on while preparing these solutions.

### Download PDF of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials                  ### Exercise 2.1 Page: 28

1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case. Solutions:

Graphical method to find zeroes:-

Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis.

1. In the given graph, the number of zeroes of p(x) is 0 because the graph is parallel to x-axis does not cut it at any point.
2. In the given graph, the number of zeroes of p(x) is 1 because the graph intersects the x-axis at only one point.
3. In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at any three points.
4. In the given graph, the number of zeroes of p(x) is 2 because the graph intersects the x-axis at two points.
5. In the given graph, the number of zeroes of p(x) is 4 because the graph intersects the x-axis at four points.
6. In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at three points.

### Class 10 Maths Chapter 2 Exercise 2.2 Page: 33

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

Solutions:  2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i)1/4,-1

Solution: (ii)√2,1/3

Solution: (iii) 0,√5 (iv)1, 1 (v)-1/4,1/4

Solution: (vi) 4, 1

Solution: ### Class 10 Maths Chapter 2 Exercise 2.3 Page: 36

2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

(i)t2-3, 2t4+3t3-2t2-9t-12 ,

Solutions: Given,

First polynomial = t2 – 3

Second polynomial = 2t4 + 3t3-2t2-9t-12 (ii) x2+3x+1, 3x4+5x3-7x2+2x+2,

Solutions: Given,

First polynomial = x2+3x+1

Second polynomial = 3x4+5x3-7x2+2x+2 (iii)x3-3x+1, x5-4x3+x2+3x+1 ,

Solutions: Given,

First polynomial = x3-3x+1

Second polynomial = x5-4x3+x2+3x+1 3. Obtain all other zeroes of, if two of its zeroes are √5/3 and -√5/3.

Solutions: Since this is a polynomial equation of degree 4, hence there will be total 4 roots. (3x2−5)=0, is a factor of given polynomial f(x).

Now, when we will divide f(x) by (3x2−5) the quotient obtained will also be a factor of f(x) and the remainder will be 0. Therefore, 3x+ 6x− 2x− 10x – 5 = (3x– 5) (x2 + 2x +1)

Now, on further factorizing (x2 + 2x +1) we get,

x2 + 2x +1 = x2 + x + x +1 = 0

x(x + 1) + 1(x+1) = 0

(x+1) (x+1) = 0

So, its zeroes are given by: x= −1 and x = −1.

Therefore, all four zeroes of given polynomial equation are:

√5/3, -√5/3, −1 and −1.

4. On dividing by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

Solutions: Given,

Dividend, p(x) = x3-3x2+x+2

Quotient = x-2

Remainder = –2x + 4

We have to find the value of Divisor, g(x) =?

As we know,

Dividend = Divisor × Quotient + Remainder

∴ x3-3x2+x+2 = g(x) × (x-2) + (–2x + 4)

x3-3x2+x+2 –(–2x + 4) = g(x) × (x-2)

Therefore, g(x) × (x-2) = x3-3x2+x-2

Now, for finding g(x) we will divide x3-3x2+x-2 with (x-2) Therefore, g(x) = (x2– x +1)

5. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) deg r(x) = 0

Solutions: According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula;

Dividend = Divisor × Quotient + Remainder

∴ p(x) = g(x) × q(x) + r(x)

Where r(x) = 0 or degree of r(x)< degree of g(x).

Now let us proof the three given cases as per division algorithm by taking examples for each.

(i): deg p(x) = deg q(x)

Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.

Let us take an example, 3 is a polynomial to be divided by 3.

So, 3= q(x)

Thus, you can see, the degree of quotient is equal to the degree of dividend.

Hence, division algorithm is satisfied here.

(ii): deg q(x) = deg r(x)

Let us take an example,p(x)= is a polynomial to be divided by g(x)=x.

So, = x = q(x)

Also, remainder, r(x) = x

Thus, you can see, the degree of quotient is equal to the degree of remainder.

Hence, division algorithm is satisfied here.

(iii): deg r(x) = 0

The degree of remainder is 0 only when the remainder left after division algorithm is constant.

Let us take an example, p(x)= is a polynomial to be divided by g(x)=x.

So, = x = q(x)

And r(x)=1

Clearly, the degree of remainder here is 0.

Hence, division algorithm is satisfied here.

### Class 10 Maths Chapter 2 Exercise 2.4 Page: 36

1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

(i) 2x3+x2– 5x + 2; 1/2,1,-2 (ii)x3-4x2+5x+2 ; 2, 1, 1 2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively. 3. If the zeroes of the polynomial x3-3x2+x+1 are a–b, a, a + b, find a and b. 4. If two zeroes of the polynomial x4-6x3-26x2+138x-35 are 2 ± √3find other zeroes. Now, if we will divide f(x) by g(x), the quotient will also be a factor of f(x) and the remainder will be 0. So, = x4-6x3-26x2+138x-35 = (x2-4x+1)(x2–2x−35)

Now, on further factorizing (x2–2x−35) we get,

x2–(7−5)x−35 = x2–7x + 5x +35 = 0

x(x − 7) + 5 (x−7) = 0

(x+5)(x−7) = 0

So, its zeroes are given by:

x= −5 and x = 7.

Therefore, all four zeroes of given polynomial equation are: 2 +√3 , 2 -√3 , −5 and 7.

 Also Access NCERT Exemplar for class 10 Maths Chapter 2 CBSE Notes for class 10 Maths Chapter 2

## NCERT Solutions for class 10 Maths Chapter 2- Polynomials

As this is one of the important topics in maths, it comes under the unit – Algebra which has a weightage of 20 marks in the class 10 maths board exams. The average number of questions asked from this chapter is usually 1.

This chapter talks about the following,

• Introduction to Polynomials
• Geometrical Meaning of the Zeros of Polynomial
• Relationship between Zeros and Coefficients of a Polynomial
• Division Algorithm for Polynomials

List of Exercises in class 10 Maths Chapter 2

Exercise 2.1 Solutions 1 Question ( 1 short)

Exercise 2.2 Solutions 2 Question ( 2 short)

Exercise 2.3 Solutions 5 Question ( 2 short, 5 long)

Exercise 2.4 Solutions 5 Question ( 1 short, 4 long)

NCERT solutions for class 10 maths chapter 2 – Polynomials is the second chapter of class 10 Maths. Polynomials are introduced in class 9 where we discussed polynomials in one variable and their degrees in the previous class and this is discussed more in details in class 10. The chapter discusses the polynomials and their applications. We study about the division algorithm for polynomials of integers and also whether the zeroes of quadratic polynomials are related to its coefficients.

The chapter starts with the introduction of polynomials in section 2.1 followed by two very important topics in section 2.2 and 2.3

• Geometrical Meaning of the zeroes of a Polynomial – It includes 1 question having 6 different cases.
• Relationship between Zeroes and Coefficients of a polynomial – Explore the relationship between zeroes and coefficients of a quadratic polynomial through solutions to 2 problems in Exercise 2.2 having 6 parts in each question.

Next, it discusses the following topics which were introduced in class 9.

• Division Algorithm for Polynomials – In this, the solutions for 5 problems in Exercise 2.3 is given having three long questions.

### Key Features of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials

• It covers the whole syllabus of Class 10 Maths.
• After studying through these NCERT solutions prepared by our subject experts, You will be confident to score well in exams.
• It follows NCERT guidelines which help in preparing the students accordingly.
• It contains all the important questions from the examination point of view.
• It helps in scoring well in maths in exams.

## Frequently Asked Questions on Chapter 2- Polynomials

### The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case?

Graphical method to find zeroes:-

Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis.

In the given graph, the number of zeroes of p(x) is 0 because the graph is parallel to x-axis does not cut it at any point.

### Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients 4u2+ 8u?

4u(u+2)

Therefore, zeroes of polynomial equation 4 + 8u are {0, -2}.

Sum of zeroes = 0+(-2) = -2 =-8/4 =(-coefficient of u)/coefficient of u2

Product of zeroes = 0x-2 = 0 = 0/4 =constant term/coefficient of u2

### Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg p(x) = deg q(x)?

According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula;

Dividend = Divisor × Quotient + Remainder

∴ p(x) = g(x) × q(x) + r(x)

Where r(x) = 0 or degree of r(x)< degree of g(x).

Now let us proof the three given cases as per division algorithm by taking examples for each.

deg p(x) = deg q(x)

Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.Let us take an example, 3x2+3x+3 is a polynomial to be divided by 3.

So, 3x2+3x+3/3=x2+x+1=q(x)

Thus, you can see, the degree of quotient is equal to the degree of dividend.

Hence, division algorithm is satisfied here.

### Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively?

Let us consider the cubic polynomial is ax3+bx2+cx+d and the values of the zeroes of the polynomials be α, β, γ.

As per the given question,

α + β + γ = -b/a = 2/1

αβ + βγ + γα = c/a = -7/1

α β γ = -d/a = -14/1

Thus, from above three expressions we get the values of coefficient of polynomial.

a = 1, b = -2, c = -7, d = 14

Hence, the cubic polynomial is x3-2x2-7x+14