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## Answers of Mathematics NCERT solutions for class 10 Chapter 2 Polynomials

**Unit 2**

**Polynomials**

**Exercise 2.1**

***Question 1:**

The graphs of _{}are given in following ﬁgure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

**Answer:**

- The number of zeroes is 0 as the graph does not cut the x-axis at any point.

- The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.

- The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

- The number of zeroes is 2 as the graph intersects the x-axis at 2 points.

- The number of zeroes is 4 as the graph intersects the x-axis at 4 points.

- The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

**Exercise 2.2**

**Question 1:**

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

**Answer:**

The value of_{}is zero when _{}i.e., when x = 4 or x = −2

Therefore, the zeroes of_{}are 4 and −2.

Sum of zeroes =

Product of zeroes

The value of _{}is zero when _{}, i.e.,_{}

Therefore, the zeroes of 4s^{2 }− 4s + 1 are_{} .

Sum of zeroes =

Product of zeroes

The value of 6x^{2 }− 3 − 7x is zero when _{}

Therefore, the zeroes of_{}

Sum of zeroes =

Product of zeroes =

The value of _{}is zero when _{} Therefore, the zeroes of _{}are 0 and −2.

Sum of zeroes =

Product of zeroes =

The value of t^{2 }− 15 is zero when_{}when

Therefore, the zeroes of t^{2 }− 15 are_{} .

Sum of zeroes =

Product of zeroes =

The value of _{}is zero when _{}_{ }when or

Therefore, the zeroes of _{}

Sum of zeroes =

Product of zeroes

***Question 2:**

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

**Answer:**

Let the polynomial be_{} and its zeroes be_{}

Therefore, the quadratic polynomial is _{}

(ii) _{}

Let the polynomial be_{} and its zeroes be_{}

Therefore, the quadratic polynomial is _{}

Let the polynomial be_{} and its zeroes be_{}

Therefore, the quadratic polynomial is_{}

Let the polynomial be_{} and its zeroes be_{}

Therefore, the quadratic polynomial is

Let the polynomial be_{} and its zeroes be_{}

Therefore, the quadratic polynomial is

Let the polynomial be_{}

Therefore, the quadratic polynomial is_{}

**Exercise 2.3**

**Question 1:**

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

**Answer:**

Quotient = x − 3 Remainder = 7x − 9

Quotient = x^{2 }+ x − 3 Remainder = 8

Quotient = −x^{2 }− 2

Remainder = −5x +10

**Question 2:**

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

**Answer:**

=

Since the remainder is 0,

Hence, is a factor of .

Since the remainder is 0,

Hence, _{} is a factor of _{}

Since the remainder _{}

Hence,_{}is not a factor of_{}

***Question 3:**

Obtain all other zeroes of_{} if two of its zeroes are _{}

Answer:

Since the two zeroes are_{},

_{}is a factor of_{}

Therefore, we divide the given polynomial by_{}

We factorize_{}

Therefore, its zero is given by _{}

As it has the term_{} therefore, there will be 2 zeroes at x = −1.

Hence, the zeroes of the given polynomial are_{}and −1.

***Question 4:**

On dividing_{}by a polynomial g(x), the quotient and remainder were x − 2 and − 2x + 4, respectively. Find g(x).

**Answer:**

g(x) = ? (Divisor)

Quotient _{}

Remainder _{}

Dividend = Divisor × Quotient + Remainder

g(x) is the quotient when we divide_{}by_{}

**Question 5:**

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

- deg p(x) = deg q(x)
- deg q(x) = deg r(x)
- deg r(x) = 0

**Answer:**

According to the division algorithm, if p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can ﬁnd polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x)

Degree of a polynomial is the highest power of the variable in the polynomial.

- deg p(x) = deg q(x)

Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).

Let us assume the division of_{} by 2.

Here, _{}

g(x) = 2

Degree of p(x) and q(x) is the same i.e., 2. Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

Thus, the division algorithm is satisﬁed.

- deg q(x) = deg r(x)

Let us assume the division of _{}

Here, _{}

Clearly, the degree of q(x) and r(x) is the same i.e., 1. Checking for division algorithm,

Thus, the division algorithm is satisﬁed. (iii)deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant. Let us assume the division of x^{3 }+ 1by x^{2}.

Here, p(x) = x^{3 }+ 1 g(x) = x^{2}

q(x) = x and r(x) = 1

Clearly, the degree of r(x) is 0. Checking for division algorithm, p(x) = g(x) × q(x) + r(x)

x^{3 }+ 1 = (x^{2 }) × x + 1 x^{3 }+ 1 = x^{3 }+ 1

Thus, the division algorithm is satisﬁed.

**Exercise 2.4**

**Question 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:

**Answer:**

(i)

Therefore,_{}are the zeroes of the given polynomial.

Comparing the given polynomial with_{}we obtain _{}

Therefore, the relationship between the zeroes and the coefficients is veriﬁed.

(ii)

Therefore, 2, 1, 1 are the zeroes of the given polynomial.

Comparing the given polynomial with _{} , we obtain a = 1, b = −4, c = 5, d =−2.

Veriﬁcation of the relationship between zeroes and coefficient of the given polynomial

Multiplication of zeroes taking two at a time _{}

Multiplication of zeroes _{}

Hence, the relationship between the zeroes and the coefficients is veriﬁed.

**Question 2:**

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

**Answer:**

Let the polynomial be _{} and the zeroes be_{}be

It is given that

If a = 1, then b = −2, c = −7, d = 14

Hence, the polynomial is_{}

**Question 3:**

If the zeroes of polynomial_{}are_{} ﬁnd a and b.

**Answer:**

Zeroes are a − b, a + a + b

Comparing the given polynomial with_{} we obtain

p = 1, q = −3, r = 1, t = 1

The zeroes are _{}

Hence, a = 1 and _{}or_{}

***Question 4:**

If two zeroes of the polynomial_{} are _{}ﬁnd other zeroes.

**Answer:**

Given that _{}and_{}are zeroes of the given polynomial.

Therefore, _{}

= x^{2 }− 4x + 1 is a factor of the given polynomial

Clearly,_{}

It can be observed that _{} is also a factor of the given polynomial.

And _{}

Therefore, the value of the polynomial is also zero when_{}

Or x = 7 or −5

Hence, 7 and −5 are also zeroes of this polynomial.

***Question 5:**

If the polynomial_{}is divided by another polynomial_{} the remainder comes out to be x + a, ﬁnd k and a.

**Answer:**

By division algorithm,

Dividend = Divisor × Quotient + Remainder

Dividend − Remainder = Divisor × Quotient

will be perfectly divisible by

.

Let us divide_{}by_{}

It can be observed that_{}will be 0.

Therefore,_{}and_{}

For_{}

And thus, k = 5

For_{}

Therefore, a = −5 Hence, k = 5 and a = −5

## 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

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 mathematics 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 – 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.

## 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 SWC’S 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 SWC’S. 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 SWC’S website.

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

The concepts are covered in NCERT Solution for Class 10 Mathematics 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.