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# NCERT Solutions for Class 10 Maths Chapter 2 : Polynomials

Polynomials class 10 Maths NCERT Solutions Chapter is concerned with identifying the zeros of given polynomials. You will also learn about the division algorithm, which is quite significant. Students preparing for class 10 maths chapter 2 will be able to clear all their concepts on polynomials at the root level. Expert faculty of Toppr produced these solutions to assist students with their first term exam preparations. It covers all major concepts in detail, allowing students to understand the ideas better.

NCERT solutions for class 10 maths chapter 2 concentrates on the essential concepts such as the geometrical significance of polynomial zeros, the link between polynomial zeros and coefficients, and the polynomial division algorithm. All of these solutions are designed with the new CBSE pattern in mind so that students have a complete understanding of their tests.

Polynomial class 10 Questions and Answers are very useful for getting good grades in tests and properly preparing you with all of the important concepts. These NCERT Solutions are valuable tools that can assist you not only in covering the full syllabus but also in providing an in-depth analysis of the subjects. The Class 10 Maths NCERT Solutions Chapter 2 are available in pdf format below, and some of them are also included in the exercises.

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## Access NCERT Solutions for Class 10 Maths Chapter 2 : Polynomials

Exercise 2.1
Question 1
The graphs of are given in the figure, for some polynomials . Find the number of zeroes of , in each case.
Medium
Solution
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(i) The graph does not intersect the -axis, so there are no zeros in .

(ii) The graph intersects the -axis at one place, so here there is one zero in .

(iii) The graph intersects the -axis at three places, so that there are three zeros in .

(iv) The graph intersects the -axis at two places, so here there are two zeros in .

(v) The graph intersects the -axis at four places, so there are four zeros in .

(vi) The graph intersects the -axis at three places, so that there are three zeros in .
Exercise 2.2
Question 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i)  (ii)  (iii)
(iv)
(v)  (vi)
Medium
Solution
Verified by Toppr
(i)

Factorize the equation, we get
So, the value of is zero when i.e., when or .
Therefore, the zeros of are -2 and 4.
Now,
Sum of zeroes =

Product of zeros =

(ii)

Factorize the equation, we get
So, the value of is zero when , i.e., when or .
Therefore, the zeros of are and .
Now,
Sum of zeroes =

Product of zeros =

(iii)

Factorize the equation, we get
So, the value of is zero when , i.e., when or .
Therefore, the zeros of are and .
Now,
Sum of zeroes =

Product of zeros =

(iv)

Factorize the equation, we get
So, the value of is zero when , i.e., when or .
Therefore, the zeros of are and
Now,
Sum of zeroes =

Product of zeros =

(v)

Factorize the equation, we get
So, the value of is zero when , i.e., when or .
Therefore, the zeros of are .
Now,
Sum of zeroes =

Product of zeros =

(vi)

Factorize the equation, we get
So, the value of is zero when x + 1 = 0, 3x - 4 = 0, i.e., when x = -1 or x = .
Therefore, the zeros of are -1 and .
Now,
Sum of zeroes =

Product of zeros =
Question 2
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) , (ii) (iii)
(iv) (v) (vi)
Medium
Solution
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(i) , -1

Substitute the value in the formula, we get

(ii)

Substitute the value in the formula, we get

Multiply by 3 to remove denominator,

(iii) 0,

Substitute the value in the formula, we get

(iv) 1, 1

Substitute the value in the formula, we get

(v)

Substitute the value in the formula, we get

Multiply by

(vi) 4, 1

Substitute the value in the formula, we get
Exercise 2.3
Question 1
Divide the polynomial by the polynomial and find the quotient and remainder.

A
B
C
D
Medium
Solution
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Dividing by , we get
Quotient and Remainder,

Hence, option D.
Question 2
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i)
(ii)
(iii)
Medium
Solution
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(i)
Remainder is 0, hence is a factor of

(ii)
Remainder is 0, hence is a factor of

(iii)
Remainder is 2, hence is not a factor of
Question 3
Obtain all other zeroes of , if two of its zeroes are and .
Medium
Solution
Verified by Toppr
Two zeroes are and

So we can write it as, x = and x =

we get and

Multiply both the factors we get,

Multiply by we get
is the factor of

Now divide,  by  we get,
Quotient is

Compare the equation with quadratic formula,

Sum of root
Product of the root

So, we get

So, our zeroes are and
Question 4
On dividing by a polynomial , the quotient and remainder were and , respectively. Find .
A
B
C
D
Medium
Solution
Verified by Toppr
Dividend
Quotient
Remainder
By division algorithm,

So,
Question 5
Give examples of polynomials and , which satisfy the division algorithm and
(i) deg deg  (ii) deg deg  (iii) deg
Medium
Solution
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(i) deg p(x) = deg q(x)

We know the formula,

Dividend = Divisor x quotient + Remainder

So here the degree of quotient will be equal to degree of dividend when the divisor is constant.

Let us assume the division of by .
Here,

and

Degree of and is the same i.e., .

Checking for division algorithm,

Hence, the division algorithm is satisfied.

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

Let us assume the division of by ,
Here, p(x) = , g(x) = , q(x) = x and r(x) = x

Degree of q(x) and r(x) is the same i.e., 1.

Checking for division algorithm,

Hence, 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 by
Here, p(x) =
g(x) =
and

Degree of is

Checking for division algorithm,

Hence, 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:
(i)

(ii)
Medium
Solution
Verified by Toppr
(i)
.... (1)

Zeroes for this polynomial are

Substitute the in equation (1)

Substitute the in equation (1)

Substitute the in equation (1)

Therefore, are the zeroes of the given polynomial.

Comparing the given polynomial with we obtain,

Let us assume , ,
Sum of the roots =

Product of the roots =

Therefore, the relationship between the zeroes and coefficient are verified.

(ii)
.... (1)
Zeroes for this polynomial are

Substitute in equation (1)

Substitute in equation (1)

Therefore, are the zeroes of the given polynomial.

Comparing the given polynomial with we obtain,

Let us assume , ,

Sum of the roots =

Multiplication of two zeroes taking two at a time=

Product of the roots =

Therefore, the relationship between the zeroes and coefficient are verified.
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### NCERT Solutions for Class 10 Maths Chapter 2 Polynomials – Brief Overview

#### Polynomials

A polynomial is an expression made up of variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, division, and a non-negative integer exponent.

Polynomials are algebraic expressions with more than two terms. They can also be described as the sum of numerous phrases in which the same variable(s) has/have varying powers.

#### Degree of a Polynomial

The degree of a polynomial in mathematics is the highest or greatest power of a variable in a polynomial equation. To determine the degree of any polynomial, just variables are evaluated; coefficients are ignored.

For example,

Equation -> x + 10    Degree -> 1

Equation -> 2x2 + x + 8    Degree -> 2

Equation -> 10x6 + 3x3 - 10    Degree -> 6

#### Standard Forms of Polynomials

1. Linear Polynomial - ax + b, where a, b are real numbers and a ≠ 0.
2. Quadratic Polynomial - ax2+ bx + c, where a, b, c are real numbers & a ≠ 0.
3. Cubic Polynomial - ax3+ bx2 + cx + d, where a, b, c, d are real numbers and a ≠ 0.

#### Division Algorithm for Polynomials

If p(x) and g(x) are any 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). This result is known as the Division Algorithm for polynomial

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### Frequently Asked Questions on NCERT Class 10 Maths Chapter 2 : Polynomials

Q1. What can be the real-life applications of polynomials?

Answer: Professionals that need to perform difficult computations are the most likely to employ polynomials on a daily basis. Polynomials are used by civil engineers to design highways, buildings, and other constructions. Polynomials can be used to forecast traffic patterns and then use them to conduct traffic management measures.

Q2. How many exercises are in NCERT Solutions for Class 10 Maths Chapter 2?

Answer: The NCERT Solutions Class 10 Maths Chapter 2 Polynomials is made up of four exercises and 13 well-researched problems. These 13 sums are well-placed in relation to the theory's context. These issues come in a variety of styles, including equations, long answer type questions, word problems, proof-based questions, and many others. On Toppr, students can discover answers to all of the problems in these four exercises.

Q3. What are the Key Topics in NCERT Solutions Class 10 Maths Chapter 2?

Answer: The concept of a polynomials zero, degrees and coefficients, and the division algorithm of polynomials are all key subjects discussed in NCERT Solutions Class 10 Maths Chapter 2. Polynomials Chapter 2 is quite diversified and aids students in understanding concepts such as parabolas, graphs, and the geometric significance of polynomial zeros. Students will be able to answer polynomial problems after understanding these topics.

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