# NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

**NCERT Solutions Class 10 Maths Chapter 2 Polynomials** are provided here to help the students in learning efficiently for their exams. 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 practise the questions of Chapter 2 Polynomials using the **Solutions of NCERT**. This makes it simple for the students to learn by adding step-wise explanations to these Maths NCERT Class 10 Solutions.

**NCERT Solutions for Class 10 Maths** is 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, following the NCERT guidelines is focused on while preparing these solutions.Chapter 2 PolynomialsExercise 2.1

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## 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.

(i) 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.

(ii) In the given graph, the number of zeroes of p(x) is 1 because the graph intersects the x-axis at only one point.

(iii) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at any three points.

(iv) In the given graph, the number of zeroes of p(x) is 2 because the graph intersects the x-axis at two points.

(v) In the given graph, the number of zeroes of p(x) is 4 because the graph intersects the x-axis at four points.

(vi) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at three points.

## 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:**

**(i) x ^{2}โ2x โ8**

**โ**x^{2}โ 4x+2xโ8 = x(xโ4)+2(xโ4) = (x-4)(x+2)

Therefore, zeroes of polynomial equation x^{2}โ2xโ8 are (4, -2)

Sum of zeroes = 4โ2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = 4ร(-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x^{2})

**(ii) 4s ^{2}โ4s+1**

โ4s^{2}โ2sโ2s+1 = 2s(2sโ1)โ1(2s-1) = (2sโ1)(2sโ1)

Therefore, zeroes of polynomial equation 4s^{2}โ4s+1 are (1/2, 1/2)

Sum of zeroes = (ยฝ)+(1/2) = 1 = -4/4 = -(Coefficient of s)/(Coefficient of s^{2})

Product of zeros = (1/2)ร(1/2) = 1/4 = (Constant term)/(Coefficient of s^{2 })

**(iii) 6x ^{2}โ3โ7x**

โ6x^{2}โ7xโ3 = 6x^{2 }โ 9x + 2x โ 3 = 3x(2x โ 3) +1(2x โ 3) = (3x+1)(2x-3)

Therefore, zeroes of polynomial equation 6x^{2}โ3โ7x are (-1/3, 3/2)

Sum of zeroes = -(1/3)+(3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = -(1/3)ร(3/2) = -(3/6) = (Constant term) /(Coefficient of x^{2 })

**(iv) 4u ^{2}+8u**

โ 4u(u+2)

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

Sum of zeroes = 0+(-2) = -2 = -(8/4) = = -(Coefficient of u)/(Coefficient of u^{2})

Product of zeroes = 0ร-2 = 0 = 0/4 = (Constant term)/(Coefficient of u^{2 })

**(v) t ^{2}โ15**

โ t^{2} = 15 or t = ยฑโ15

Therefore, zeroes of polynomial equation t^{2} โ15 are (โ15, -โ15)

Sum of zeroes =โ15+(-โ15) = 0= -(0/1)= -(Coefficient of t) / (Coefficient of t^{2})

Product of zeroes = โ15ร(-โ15) = -15 = -15/1 = (Constant term) / (Coefficient of t^{2 })

**(vi) 3x ^{2}โxโ4**

โ 3x^{2}โ4x+3xโ4 = x(3x-4)+1(3x-4) = (3x โ 4)(x + 1)

Therefore, zeroes of polynomial equation3x^{2} โ x โ 4 are (4/3, -1)

Sum of zeroes = (4/3)+(-1) = (1/3)= -(-1/3) = -(Coefficient of x) / (Coefficient of x^{2})

Product of zeroes=(4/3)ร(-1) = (-4/3) = (Constant term) /(Coefficient of x^{2 })

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

**(i) 1/4 , -1**

**Solution:**

From the formulas of sum and product of zeroes, we know,

Sum of zeroes = ฮฑ+ฮฒ

Product of zeroes = ฮฑ ฮฒ

Sum of zeroes = ฮฑ+ฮฒ = 1/4

Product of zeroes = ฮฑ ฮฒ = -1

โด If ฮฑ and ฮฒ are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

**x ^{2}โ(ฮฑ+ฮฒ)x +ฮฑฮฒ = 0**

**x ^{2}โ(1/4)x +(-1) = 0**

**4x ^{2}โx-4 = 0**

**Thus,4x ^{2}โxโ4 is the **quadratic polynomial.

**(ii)**โ2, 1/3

**Solution:**

Sum of zeroes = ฮฑ + ฮฒ =โ2

Product of zeroes = ฮฑ ฮฒ = 1/3

โด If ฮฑ and ฮฒ are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

**x ^{2}โ(ฮฑ+ฮฒ)x +ฮฑฮฒ = 0**

**x ^{2} โ(**โ2

**)x + (1/3) = 0**

**3x ^{2}-3**โ2x+1 = 0

**Thus, 3x ^{2}-3**โ2x+1

**is the**quadratic polynomial.

**(iii) 0, โ5**

**Solution:**

Given,

Sum of zeroes = ฮฑ+ฮฒ = 0

Product of zeroes = ฮฑ ฮฒ = โ5

โด If ฮฑ and ฮฒ are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly

as:-

**x ^{2}โ(ฮฑ+ฮฒ)x +ฮฑฮฒ = 0**

**x ^{2}โ(0)x +**โ5

**= 0**

**Thus, x ^{2}+**โ5

**is the**quadratic polynomial.

**(iv) 1, 1**

**Solution:**

Given,

Sum of zeroes = ฮฑ+ฮฒ = 1

Product of zeroes = ฮฑ ฮฒ = 1

โด If ฮฑ and ฮฒ are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

**x ^{2}โ(ฮฑ+ฮฒ)x +ฮฑฮฒ = 0**

**x ^{2}โx+1 = 0**

**Thus , x ^{2}โx+1is the **quadratic polynomial.

**(v) -1/4, 1/4**

**Solution:**

Given,

Sum of zeroes = ฮฑ+ฮฒ = -1/4

Product of zeroes = ฮฑ ฮฒ = 1/4

**x ^{2}โ(ฮฑ+ฮฒ)x +ฮฑฮฒ = 0**

**x ^{2}โ(-1/4)x +(1/4) = 0**

**4x ^{2}+x+1 = 0**

**Thus,4x ^{2}+x+1 is the **quadratic polynomial.

**(vi) 4, 1**

**Solution:**

Given,

Sum of zeroes = ฮฑ+ฮฒ =

Product of zeroes = ฮฑฮฒ = 1

**x ^{2}โ(ฮฑ+ฮฒ)x+ฮฑฮฒ = 0**

**x ^{2}โ4x+1 = 0**

**Thus, x ^{2}โ4x+1 is the **quadratic polynomial.

## Exercise 2.3 Page: 36

**1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:**

**(i)** **p(x) = x ^{3}-3x^{2}+5xโ3 , g(x) = x^{2}โ2**

**Solution:**

Given,

Dividend = p(x) = x^{3}-3x^{2}+5xโ3

Divisor = g(x) = x^{2}โ 2

Therefore, upon division we get,

Quotient = xโ3

Remainder = 7xโ9

**(ii) p(x) = x ^{4}-3x^{2}+4x+5 , g(x) = x^{2}+1-x**

**Solution:**

Given,

Dividend = p(x) = x^{4 }โ 3x^{2 }+ 4x +5

Divisor = g(x) = x^{2} +1-x

Therefore, upon division we get,

Quotient = x^{2 }+ xโ3

Remainder = 8

**(iii) p(x) =x ^{4}โ5x+6, g(x) = 2โx^{2}**

**Solution:**

Given,

Dividend = p(x) =x^{4} โ 5x + 6 = x^{4 }+0x^{2}โ5x+6

Divisor = g(x) = 2โx^{2} = โx^{2}+2

Therefore, upon division we get,

Quotient = -x^{2}-2

Remainder = -5x + 10

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

**(i) t ^{2}-3, 2t^{4 }+3t^{3}-2t^{2}-9t-12**

**Solutions:**

Given,

First polynomial = t^{2}-3

Second polynomial = 2t^{4 }+3t^{3}-2t^{2 }-9t-12

As we can see, the remainder is left as 0. Therefore, we say that, t^{2}-3 is a factor of 2t^{2}+3t+4.

**(ii)x ^{2}+3x+1 , 3x^{4}+5x^{3}-7x^{2}+2x+2**

**Solutions:**

Given,

First polynomial = x^{2}+3x+1

Second polynomial = 3x^{4}+5x^{3}-7x^{2}+2x+2

As we can see, the remainder is left as 0. Therefore, we say that, x^{2} + 3x + 1 is a factor of 3x^{4}+5x^{3}-7x^{2}+2x+2.

**(iii) x ^{3}-3x+1, x^{5}-4x^{3}+x^{2}+3x+1**

**Solutions:**

Given,

First polynomial = x^{3}-3x+1

Second polynomial = x^{5}-4x^{3}+x^{2}+3x+1

As we can see, the remainder is not equal to 0. Therefore, we say that, x^{3}-3x+1 is not a factor of x^{5}-4x^{3}+x^{2}+3x+1 .

**3. Obtain all other zeroes of 3x ^{4}+6x^{3}-2x^{2}-10x-5, if two of its zeroes are โ(5/3) and โ โ(5/3).**

**Solutions:**

Since this is a polynomial equation of degree 4, hence there will be total 4 roots.

**โ(5/3) and โ โ(5/3) **are zeroes of polynomial f(x).

**โด **(x โ**โ(5/3)**) (x+**โ(5/3) **= x^{2}-(5/3) = 0

**(3x ^{2}โ5)=0,** is a factor of given polynomial f(x).

Now, when we will divide f(x) by (3x^{2}โ5) the quotient obtained will also be a factor of f(x) and the remainder will be 0.

Therefore, 3x^{4 }+6x^{3 }โ2x^{2 }โ10xโ5 = (3x^{2 }โ5)**(x ^{2}+2x+1)**

Now, on further factorizing (x^{2}+2x+1) we get,

**x ^{2}+2x+1** = x

^{2}+x+x+1 = 0

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

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

So, its zeroes are given by: **x= โ1 **and** x = โ1.**

Therefore, all four zeroes of given polynomial equation are:

**โ(5/3),- โ(5/3) , โ1 and โ1.**

Hence, is the answer.

**4. On dividing x ^{3}-3x^{2}+x+2**

**by a polynomial g(x), the quotient and remainder were xโ2 and โ2x+4, respectively. Find g(x).**

**Solution:**

Given,

Dividend, p(x) = x^{3}-3x^{2}+x+2

Quotient = x-2

Remainder = โ2x+4

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

As we know,

Dividend = Divisor ร Quotient + Remainder

โด x^{3}-3x^{2}+x+2 = g(x)ร(x-2) + (-2x+4)

x^{3}-3x^{2}+x+2-(-2x+4) = g(x)ร(x-2)

Therefore, g(x) ร (x-2) = x^{3}-3x^{2}+x+2

Now, for finding g(x) we will divide x^{3}-3x^{2}+x+2 with (x-2)

Therefore, **g(x) = (x ^{2}โx+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)โ 0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula;

Dividend = Divisor ร Quotient + Remainder

โด p(x) = g(x)รq(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, p(x) = 3x^{2}+3x+3 is a polynomial to be divided by g(x) = 3.

So, (3x^{2}+3x+3)/3 = x^{2}+x+1 = q(x)

Thus, you can see, the degree of quotient q(x) = 2, which also equal to the degree of dividend p(x).

Hence, division algorithm is satisfied here.

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

Let us take an example, p(x) = x^{2 }+ 3 is a polynomial to be divided by g(x) = x โ 1.

So, x^{2 }+ 3 = (x โ 1)ร(x) + (x + 3)

Hence, quotient q(x) = x

Also, remainder r(x) = x + 3

Thus, you can see, the degree of quotient q(x) = 1, which is also equal to the degree of remainder r(x).

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) = x^{2 }+ 1 is a polynomial to be divided by g(x) = x.

So, x^{2 }+ 1 = (x)ร(x) + 1

Hence, quotient q(x) = x

And, remainder r(x) = 1

Clearly, the degree of remainder here is 0.

Hence, division algorithm is satisfied here.

## 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) 2x ^{3}+x^{2}-5x+2; -1/2, 1, -2**

**Solution:**

Given, p(x) **= **2x^{3}+x^{2}-5x+2

And zeroes for p(x) are = 1/2, 1, -2

โด p(1/2) = 2(1/2)^{3}+(1/2)^{2}-5(1/2)+2 = (1/4)+(1/4)-(5/2)+2 = 0

p(1) = 2(1)^{3}+(1)^{2}-5(1)+2 = 0

p(-2) = 2(-2)^{3}+(-2)^{2}-5(-2)+2 = 0

Hence, proved 1/2, 1, -2 are the zeroes of 2x^{3}+x^{2}-5x+2.

Now, comparing the given polynomial with general expression, we get;

โด ax^{3}+bx^{2}+cx+d = 2x^{3}+x^{2}-5x+2

a=2, b=1, c= -5 and d = 2

As we know, if ฮฑ, ฮฒ, ฮณ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;

ฮฑ +ฮฒ+ฮณ = โb/a

ฮฑฮฒ+ฮฒฮณ+ฮณฮฑ = c/a

ฮฑ ฮฒฮณ = โ d/a.

Therefore, putting the values of zeroes of the polynomial,

ฮฑ+ฮฒ+ฮณ = ยฝ+1+(-2) = -1/2 = โb/a

ฮฑฮฒ+ฮฒฮณ+ฮณฮฑ = (1/2ร1)+(1 ร-2)+(-2ร1/2) = -5/2 = c/a

ฮฑ ฮฒ ฮณ = ยฝร1ร(-2) = -2/2 = -d/a

Hence, the relationship between the zeroes and the coefficients are satisfied.

**(ii) x ^{3}-4x^{2}+5x-2** ;

**2, 1, 1**

**Solution:**

Given, p(x) = x^{3}-4x^{2}+5x-2

And zeroes for p(x) are 2,1,1.

โด p(2)= 2^{3}-4(2)^{2}+5(2)-2 = 0

p(1) = 1^{3}-(4ร1^{2 })+(5ร1)-2 = 0

Hence proved, 2, 1, 1 are the zeroes of x^{3}-4x^{2}+5x-2

Now, comparing the given polynomial with general expression, we get;

โด ax^{3}+bx^{2}+cx+d = x^{3}-4x^{2}+5x-2

a = 1, b = -4, c = 5 and d = -2

As we know, if ฮฑ, ฮฒ, ฮณ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;

ฮฑ + ฮฒ + ฮณ = โb/a

ฮฑฮฒ + ฮฒฮณ + ฮณฮฑ = c/a

ฮฑ ฮฒ ฮณ = โ d/a.

Therefore, putting the values of zeroes of the polynomial,

ฮฑ +ฮฒ+ฮณ = 2+1+1 = 4 = -(-4)/1 = โb/a

ฮฑฮฒ+ฮฒฮณ+ฮณฮฑ = 2ร1+1ร1+1ร2 = 5 = 5/1= c/a

ฮฑฮฒฮณ = 2ร1ร1 = 2 = -(-2)/1 = -d/a

Hence, the relationship between the zeroes and the coefficients are satisfied.

**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, โ7, โ14 respectively.**

**Solution:**

Let us consider the cubic polynomial is ax^{3}+bx^{2}+cx+d and the values of the zeroes of the polynomials be ฮฑ, ฮฒ, ฮณ.

As per the given question,

ฮฑ+ฮฒ+ฮณ = -b/a = 2/1

ฮฑฮฒ +ฮฒฮณ+ฮณฮฑ = c/a = -7/1

ฮฑ ฮฒฮณ = -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 x^{3}-2x^{2}-7x+14

**3. If the zeroes of the polynomial x ^{3}-3x^{2}+x+1**

**are a โ b, a, a + b, find a and b.**

**Solution:**

We are given with the polynomial here,

p(x) = x^{3}-3x^{2}+x+1

And zeroes are given as a โ b, a, a + b

Now, comparing the given polynomial with general expression, we get;

โดpx^{3}+qx^{2}+rx+s = x^{3}-3x^{2}+x+1

p = 1, q = -3, r = 1 and s = 1

Sum of zeroes = a โ b + a + a + b

-q/p = 3a

Putting the values q and p.

-(-3)/1 = 3a

a=1

Thus, the zeroes are 1-b, 1, 1+b.

Now, product of zeroes = 1(1-b)(1+b)

-s/p = 1-b^{2}

-1/1 = 1-b^{2}

b^{2} = 1+1 = 2

b = โ2

Hence,1-โ2, 1 ,1+โ2 are the zeroes of x^{3}-3x^{2}+x+1.

**4. If two zeroes of the polynomial x ^{4}-6x^{3}-26x^{2}+138x-35**

**are 2 ยฑ**โ

**3,**

**find other zeroes.**

**Solution:**

Since this is a polynomial equation of degree 4, hence there will be total 4 roots.

Let f(x) = x^{4}-6x^{3}-26x^{2}+138x-35

Since 2 +โ**3 **and 2-โ**3 **are zeroes of given polynomial f(x).

โด [xโ(2+โ**3**)] [xโ(2-โ**3)**] = 0

(xโ2โโ**3**)(xโ2+โ**3**) = 0

On multiplying the above equation we get,

x^{2}-4x+1, this is a factor of a given polynomial f(x).

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, x^{4}-6x^{3}-26x^{2}+138x-35 = (x^{2}-4x+1)(x^{2} โ2xโ35)

Now, on further factorizing (x^{2}โ2xโ35) we get,

x^{2}**โ(7โ5)x โ35** = x^{2}โ 7x+5x+35 = 0

x(x โ7)+5(xโ7) = 0

**(x+5)(xโ7) = 0**

So, its zeroes are given by:

x= โ5 and x = 7.

Therefore, all four zeroes of given polynomial equation are: 2+โ**3** , 2-โ**3**, **โ5 and 7.**

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

As this is one of the important topics in maths, it comes under the unit โ 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

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 detail in Class 10. The NCERT solutions for class 10 maths for this chapter discusses the answers for various types of questions related to polynomials and their applications. We study the division algorithm for polynomials of integers and also whether the zeroes of quadratic polynomials are related to their 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 โ It includes 1 question having 6 different cases.
- Relationship between Zeroes and Coefficients of a polynomial โ 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 โ 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.

## Frequently Asked Questions on NCERT Solutions for Class 10 Maths Chapter 2

### Where can I get the accurate solution for NCERT Solution for Class 10 Maths Chapter 2?

At BYJUโS you can get the accurate solution in PDF format for NCERT Solution for Class 10 Maths Chapter 2. The NCERT Textbook Solutions for the chapter Polynomials have been designed accurately by Mathematics experts at BYJUโS. All these solutions are provided by considering the new pattern of CBSE, so that students can get thorough knowledge for their exams.

### Is it necessary to solve each problem provided in the NCERT Solution for Class 10 Maths Chapter 2?

Yes. Because these questions are important from an exam perspective. These questions are solved by experts to help the students to crack exercise very easily. These solutions help students to familiarize themselves with the polynomials. Solutions are available in PDF format on BYJUโS website.

### List out the concepts covered in NCERT Solution for Class 10 Maths Polynomials?

The concepts are covered in NCERT Solution for Class 10 Maths Polynomials are introduction to polynomials, geometrical meaning of the zeros of polynomial, relationship between zeros and coefficients of a polynomial and division algorithm for polynomials. By learning these concepts students will be able to solve questions on polynomials.