NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials help students understand the important concepts of zeroes of a polynomial, graphs of polynomials, and the relationship between zeroes and coefficients of a quadratic polynomial. These concepts form an important part of algebra and help students develop a strong foundation for solving polynomial-based questions.

In Class 10 Maths Chapter 2 Polynomials, students learn how to identify the zeroes of a polynomial, understand their geometrical meaning using graphs, find the zeroes of quadratic polynomials, and verify the relationship between the zeroes and their coefficients. Students also learn how to form a quadratic polynomial when the sum and product of its zeroes are given.

The NCERT Solutions for Class 10 Maths Chapter 2 Polynomials provide clear explanations and step-by-step solutions to the questions given in the NCERT textbook. The solutions for Exercise 2.1 and Exercise 2.2 are explained in a simple way so that students can understand the concepts, practise different types of questions, and prepare effectively for their exams.

Students can use these Polynomials Class 10 NCERT Solutions for homework, revision, exam preparation, and clearing doubts related to zeroes of polynomials, polynomial graphs, sum and product of zeroes, and the relationship between zeroes and coefficients.

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Free PDF

Students can download the NCERT Solutions for Class 10 Maths Chapter 2 Polynomials PDF for easy access to complete solutions and explanations. The PDF covers the important concepts and questions from Class 10 Maths Chapter 2 Exercise 2.1 and Exercise 2.2, with step-by-step methods to make problem-solving easier.

Download the NCERT Solutions for Class 10 Maths Chapter 2 Polynomials PDF below and use it for quick revision, practice, and exam preparation. It can help you revise important Polynomials Class 10 formulas, understand the geometrical meaning of zeroes of a polynomial, and learn the relationship between zeroes and coefficients of quadratic polynomials.

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: All Exercises

The NCERT Solutions for Class 10 Maths Chapter 2 Polynomials cover all questions from Exercise 2.1 and Exercise 2.2 of the current NCERT textbook. These exercises focus on important concepts such as the geometrical meaning of zeroes of a polynomial, finding the zeroes of quadratic polynomials, understanding the relationship between zeroes and coefficients, and forming a quadratic polynomial when the sum and product of its zeroes are given.

The Polynomials Class 10 NCERT Solutions provide simple, step-by-step methods for solving each question. These solutions help students understand the concepts clearly, practise important question types, and prepare effectively for school and board examinations.

Class 10 Maths Chapter 2 Exercise 2.1

Class 10 Maths Chapter 2 Exercise 2.1 is based on the geometrical meaning of zeroes of a polynomial. Students are given different polynomial graphs and need to determine the number of zeroes by observing the points where each graph intersects or touches the x-axis.

A value x = r is called a zero of a polynomial if:

p(r) = 0

In graphical form, the zeroes of a polynomial are represented by the x-coordinates of the points where its graph meets the x-axis.

Class 10 Maths Chapter 2 Exercise 2.2

Class 10 Maths Chapter 2 Exercise 2.2 focuses on finding the zeroes of quadratic polynomials and verifying the relationship between zeroes and coefficients.

The general form of a quadratic polynomial is:

ax² + bx + c

If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then:

Sum of zeroes:

α + β = -b/a

Product of zeroes:

αβ = c/a

Students also learn how to form a quadratic polynomial when the sum and product of its zeroes are given.

If:

S = Sum of zeroes

and

P = Product of zeroes

then a quadratic polynomial can be written as:

x² - Sx + P

or

x² - (Sum of zeroes)x + Product of zeroes

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Overview

The NCERT Solutions for Class 10 Maths Chapter 2 help students develop a clear understanding of polynomials and their zeroes. The chapter explains how zeroes can be identified from graphs and how the zeroes of a quadratic polynomial are related to its coefficients.

A polynomial is an algebraic expression made up of variables, coefficients, and non-negative integer powers of variables. Depending on the highest power of the variable, polynomials can be classified as linear, quadratic, cubic, and higher-degree polynomials.

For example:

Linear polynomial: ax + b

Quadratic polynomial: ax² + bx + c

Cubic polynomial: ax³ + bx² + cx + d

One of the most important concepts in Class 10 Maths Chapter 2 Polynomials is the relationship between the zeroes and coefficients of a quadratic polynomial.

For the quadratic polynomial:

ax² + bx + c

if α and β are its zeroes, then:

α + β = -b/a

and

αβ = c/a

These formulas help students verify the zeroes of a quadratic polynomial and also form a new quadratic polynomial when its zeroes or their sum and product are known.

The NCERT Solutions for Class 10 Maths Chapter 2 Polynomials explain these concepts through solved examples and detailed exercise solutions. Students can use them for homework, revision, concept clarification, and exam preparation.

Important Topics in Class 10 Maths Chapter 2 Polynomials

Class 10 Maths Chapter 2 Polynomials introduces students to important concepts related to polynomial expressions, their zeroes, graphs, and coefficients. Understanding these topics makes it easier to solve questions from Class 10 Maths Exercise 2.1 and Exercise 2.2.

1. Definition of a Polynomial

A polynomial is an algebraic expression containing variables, coefficients, and powers of variables that are non-negative integers.

For example:

p(x) = 2x² + 5x - 3

Here, 2, 5, and -3 are coefficients, while the highest power of x is 2.

2. Types of Polynomials

Polynomials can be classified according to their degree.

Linear polynomial: ax + b

Degree = 1

Quadratic polynomial: ax² + bx + c

Degree = 2

Cubic polynomial: ax³ + bx² + cx + d

Degree = 3

3. Zeroes of a Polynomial

A number r is called a zero of a polynomial p(x) when substituting r in the polynomial gives zero.

The condition is:

p(r) = 0

For example, if p(2) = 0, then 2 is a zero of the polynomial.

4. Geometrical Meaning of Zeroes of a Polynomial

The zeroes of a polynomial are represented graphically by the points where the graph of the polynomial intersects or touches the x-axis.

Therefore:

Number of zeroes = Number of points where the graph meets the x-axis

This concept is especially important for solving Class 10 Maths Chapter 2 Exercise 2.1.

5. Graphs of Polynomials

Graphs help students understand the number of real zeroes of a polynomial.

A linear polynomial can have at most 1 zero.

A quadratic polynomial can have at most 2 zeroes.

A cubic polynomial can have at most 3 zeroes.

The number of real zeroes depends on how many times the graph intersects or touches the x-axis.

6. Zeroes of a Quadratic Polynomial

The general form of a quadratic polynomial is:

ax² + bx + c

The zeroes of a quadratic polynomial can often be found by factorising the polynomial and setting each factor equal to zero.

If α and β are the zeroes, their relationship with the coefficients can then be verified using the standard formulas.

7. Relationship Between Zeroes and Coefficients of a Quadratic Polynomial

For a quadratic polynomial:

ax² + bx + c

if α and β are its zeroes, then:

Sum of zeroes:

α + β = -b/a

Product of zeroes:

αβ = c/a

These are among the most important Polynomials Class 10 formulas and are frequently used in Exercise 2.2.

8. Forming a Quadratic Polynomial from Sum and Product of Zeroes

If the sum and product of the zeroes are known, a quadratic polynomial can be formed directly.

Let:

S = Sum of zeroes

P = Product of zeroes

Then:

Quadratic polynomial = x² - Sx + P

It can also be written as:

x² - (Sum of zeroes)x + Product of zeroes

For example, if the sum of the zeroes is 5 and their product is 6, then:

x² - 5x + 6

is a quadratic polynomial having those zeroes.

Important Formulas for Class 10 Maths Chapter 2 Polynomials

Students should remember the following formulas while solving questions from NCERT Solutions Class 10 Maths Chapter 2 Polynomials:

General form of a quadratic polynomial:

ax² + bx + c

Condition for r to be a zero of a polynomial:

p(r) = 0

Sum of zeroes of a quadratic polynomial:

α + β = -b/a

Product of zeroes of a quadratic polynomial:

αβ = c/a

Quadratic polynomial from sum and product of zeroes:

x² - Sx + P

Where:

S = Sum of zeroes
P = Product of zeroes

These formulas are useful for solving questions related to zeroes of quadratic polynomials, the relationship between zeroes and coefficients, and forming quadratic polynomials from their zeroes.

General Outline of Class 10 Maths Chapter 2 Polynomials

Class 10 Maths Chapter 2 Polynomials focuses mainly on understanding polynomial expressions, their zeroes, graphs, and the relationship between zeroes and coefficients of a quadratic polynomial. Students also learn how to form a quadratic polynomial when the sum and product of its zeroes are given.

The NCERT Solutions for Class 10 Maths Chapter 2 Polynomials provide step-by-step explanations for these concepts and help students solve questions from Exercise 2.1 and Exercise 2.2 with better understanding.

Introduction to Polynomials

A polynomial is an algebraic expression made up of variables, constants, and non-negative integer powers of variables. The terms of a polynomial are connected using addition, subtraction, or multiplication.

A polynomial in one variable x can be written in the general form:

p(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ

where:

• a₀, a₁, a₂, ... , aₙ are real numbers called coefficients.
• n is a non-negative integer.
• aₙ is not equal to 0.
• The highest power of x represents the degree of the polynomial.

For example:

p(x) = 3x² + 5x - 2

is a quadratic polynomial because the highest power of x is 2.

Types of Polynomials Based on Number of Terms

Polynomials can be classified according to the number of terms they contain.

Monomial: A polynomial containing only one term.

Example:

5x²

Binomial: A polynomial containing two terms.

Example:

x + 3

Trinomial: A polynomial containing three terms.

Example:

x² + 3x + 2

Types of Polynomials Based on Degree

Polynomials can also be classified according to their degree.

Linear Polynomial

General form:

ax + b

Degree = 1

Quadratic Polynomial

General form:

ax² + bx + c

Degree = 2

Cubic Polynomial

General form:

ax³ + bx² + cx + d

Degree = 3

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable having a non-zero coefficient.

For example:

For:

p(x) = 2x² + 5x - 3

the highest power of x is 2.

Therefore:

Degree = 2

For:

p(x) = 3x⁴ + 2x² - 7

the highest power of x is 4.

Therefore:

Degree = 4

Understanding the degree of a polynomial helps students identify whether a polynomial is linear, quadratic, cubic, or of a higher degree.

Zeroes of a Polynomial

One of the most important concepts in Polynomials Class 10 is finding the zeroes of a polynomial.

A number r is called a zero of a polynomial p(x) if:

p(r) = 0

In simple words, if substituting a particular value of x into the polynomial gives 0, that value is called a zero or root of the polynomial.

For example:

Consider:

p(x) = x² - 5x + 6

Factorising:

p(x) = (x - 2)(x - 3)

Therefore:

x = 2 and x = 3

are the zeroes of the polynomial.

Geometrical Meaning of Zeroes of a Polynomial

The geometrical meaning of zeroes of a polynomial can be understood through its graph.

The zeroes of a polynomial are the x-coordinates of the points where the graph of:

y = p(x)

intersects or touches the x-axis.

Therefore:

Number of zeroes = Number of points where the graph meets the x-axis

For example:

• If the graph does not meet the x-axis, it has 0 real zeroes.
• If the graph meets the x-axis at one point, it has 1 real zero.
• If the graph meets the x-axis at two points, it has 2 real zeroes.

This concept forms the basis of Class 10 Maths Chapter 2 Exercise 2.1.

Relationship Between Zeroes and Coefficients of a Quadratic Polynomial

For a quadratic polynomial:

ax² + bx + c

if α and β are its zeroes, then:

Sum of zeroes:

α + β = -b/a

Product of zeroes:

αβ = c/a

These formulas are among the most important Polynomials Class 10 formulas and are used extensively in Class 10 Maths Chapter 2 Exercise 2.2.

Forming a Quadratic Polynomial from Sum and Product of Zeroes

If the sum and product of the zeroes are given, students can form a quadratic polynomial using the formula:

x² - (Sum of zeroes)x + Product of zeroes

If:

S = Sum of zeroes

and

P = Product of zeroes

then:

Required quadratic polynomial = x² - Sx + P

For example, if:

Sum of zeroes = 5

and

Product of zeroes = 6

then:

Required polynomial = x² - 5x + 6

Any non-zero multiple of this polynomial will also have the same zeroes.

Sample NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

The following examples show the type of questions students solve in NCERT Solutions Class 10 Maths Chapter 2 Polynomials.

Exercise 2.1: Finding the Number of Zeroes from Graphs

In Class 10 Maths Chapter 2 Exercise 2.1, students observe graphs of different polynomials and determine their number of zeroes.

The basic rule is:

Number of zeroes = Number of points where the graph intersects or touches the x-axis

(i)

The graph of y = p(x) does not meet the x-axis.

Therefore:

Number of zeroes = 0

(ii)

The graph of y = p(x) meets the x-axis at one point.

Therefore:

Number of zeroes = 1

(iii)

The graph meets the x-axis at three different points.

Therefore:

Number of zeroes = 3

(iv)

The graph meets the x-axis at two points.

Therefore:

Number of zeroes = 2

(v)

The graph meets the x-axis at four points.

Therefore:

Number of zeroes = 4

(vi)

The graph meets or touches the x-axis at three points.

Therefore:

Number of zeroes = 3

Thus, the number of zeroes in the six graphs is:

0, 1, 3, 2, 4, 3

Class 10 Maths Chapter 2 Exercise 2.2: Find the Zeroes and Verify the Relationship with Coefficients

For a quadratic polynomial:

ax² + bx + c

if α and β are its zeroes, then:

α + β = -b/a

and

αβ = c/a

Question 1 (i): x² - 2x - 8

Given:

x² - 2x - 8

Factorising by splitting the middle term:

x² - 4x + 2x - 8

= x(x - 4) + 2(x - 4)

= (x + 2)(x - 4)

Therefore, the zeroes are:

α = -2

β = 4

Verification of Sum of Zeroes

α + β = -2 + 4 = 2

Using the formula:

-b/a = -(-2)/1 = 2

Therefore:

α + β = -b/a

Verification of Product of Zeroes

αβ = (-2)(4) = -8

Using the formula:

c/a = -8/1 = -8

Therefore:

αβ = c/a

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

Question 1 (ii): 4s² - 4s + 1

Given:

4s² - 4s + 1

Factorising:

4s² - 4s + 1 = (2s - 1)²

Therefore:

2s - 1 = 0

s = 1/2

The two zeroes are:

α = 1/2

β = 1/2

Sum of Zeroes

α + β = 1/2 + 1/2 = 1

Using the formula:

-b/a = -(-4)/4 = 1

Therefore:

α + β = -b/a

Product of Zeroes

αβ = (1/2)(1/2) = 1/4

Using the formula:

c/a = 1/4

Therefore:

αβ = c/a

Hence, the relationship is verified.

Question 1 (iii): 6x² - 7x - 3

Given:

6x² - 7x - 3

Splitting the middle term:

6x² - 9x + 2x - 3

= 3x(2x - 3) + 1(2x - 3)

= (2x - 3)(3x + 1)

Therefore:

2x - 3 = 0

x = 3/2

and:

3x + 1 = 0

x = -1/3

The zeroes are:

α = 3/2

β = -1/3

Sum of Zeroes

α + β = 3/2 - 1/3

= 9/6 - 2/6

= 7/6

Using:

-b/a = -(-7)/6 = 7/6

Therefore:

α + β = -b/a

Product of Zeroes

αβ = (3/2)(-1/3)

= -1/2

Using:

c/a = -3/6 = -1/2

Therefore:

αβ = c/a

Hence, the relationship is verified.

Question 1 (iv): 4u² + 8u

Given:

4u² + 8u

Taking the common factor:

4u(u + 2)

Therefore:

4u = 0

or:

u + 2 = 0

The zeroes are:

α = 0

β = -2

Sum of Zeroes

α + β = 0 - 2 = -2

Using:

-b/a = -8/4 = -2

Therefore:

α + β = -b/a

Product of Zeroes

αβ = 0 × (-2) = 0

Using:

c/a = 0/4 = 0

Therefore:

αβ = c/a

Hence, the relationship is verified.

Question 1 (v): t² - 15

Given:

t² - 15

Factorising:

t² - 15 = (t - √15)(t + √15)

Therefore, the zeroes are:

α = √15

β = -√15

Sum of Zeroes

α + β = √15 - √15 = 0

Using:

-b/a = 0

Therefore:

α + β = -b/a

Product of Zeroes

αβ = (√15)(-√15)

= -15

Using:

c/a = -15/1 = -15

Therefore:

αβ = c/a

Hence, the relationship is verified.

Question 1 (vi): 3x² - x - 4

Given:

3x² - x - 4

Splitting the middle term:

3x² - 4x + 3x - 4

= x(3x - 4) + 1(3x - 4)

= (3x - 4)(x + 1)

Therefore:

3x - 4 = 0

x = 4/3

and:

x + 1 = 0

x = -1

The zeroes are:

α = 4/3

β = -1

Sum of Zeroes

α + β = 4/3 - 1

= 1/3

Using:

-b/a = -(-1)/3 = 1/3

Therefore:

α + β = -b/a

Product of Zeroes

αβ = (4/3)(-1)

= -4/3

Using:

c/a = -4/3

Therefore:

αβ = c/a

Hence, the relationship between zeroes and coefficients is verified.

Finding a Quadratic Polynomial from the Sum and Product of its Zeroes

If:

Sum of zeroes = S

and:

Product of zeroes = P

then one quadratic polynomial is:

x² - Sx + P

(i) Sum = 1/4, Product = -1

Using:

x² - Sx + P

we get:

x² - (1/4)x - 1

Multiplying by 4:

4x² - x - 4

Therefore, one required quadratic polynomial is:

4x² - x - 4

(ii) Sum = √2, Product = 1/3

Using:

x² - Sx + P

we get:

x² - √2x + 1/3

Multiplying by 3:

3x² - 3√2x + 1

Therefore, one required quadratic polynomial is:

3x² - 3√2x + 1

(iii) Sum = 0, Product = √5

Using:

x² - Sx + P

we get:

x² - 0x + √5

Therefore:

Required polynomial = x² + √5

(iv) Sum = 1, Product = 1

Using:

x² - Sx + P

we get:

x² - x + 1

Therefore:

Required polynomial = x² - x + 1

(v) Sum = -1/4, Product = 1/4

Using:

x² - Sx + P

we get:

x² + (1/4)x + 1/4

Multiplying by 4:

4x² + x + 1

Therefore, one required quadratic polynomial is:

4x² + x + 1

(vi) Sum = 4, Product = 1

Using:

x² - Sx + P

we get:

x² - 4x + 1

Therefore:

Required polynomial = x² - 4x + 1

Benefits of Studying Class 10 Maths Chapter 2 Polynomials

Understanding Class 10 Maths Chapter 2 Polynomials is important because the concepts introduced in this chapter strengthen students' algebraic thinking and prepare them for more advanced mathematics.

1. Builds a Strong Foundation in Algebra

Polynomials are an important part of algebra. Learning about polynomial expressions, zeroes, coefficients, and graphs helps students understand related topics more easily.

2. Improves Problem-Solving Skills

Questions based on finding zeroes, factorisation, graphs, and relationships between coefficients require logical thinking. Regular practice helps students develop better problem-solving skills.

3. Helps in Understanding Quadratic Equations

The concepts of quadratic polynomials, zeroes, and factorisation are closely connected with Quadratic Equations, which students study in another Class 10 Maths chapter.

4. Develops Graph Interpretation Skills

The geometrical meaning of zeroes of a polynomial teaches students how algebraic expressions are represented graphically and how to interpret intersections with the x-axis.

5. Strengthens Understanding of Mathematical Relationships

The formulas:

α + β = -b/a

and

αβ = c/a

help students understand how the zeroes of a quadratic polynomial are connected to its coefficients.

6. Supports Exam Preparation

Questions based on zeroes of polynomials, graphs, and the relationship between zeroes and coefficients are important for practising the concepts covered in the chapter.

7. Prepares Students for Higher Mathematics

A clear understanding of polynomials supports future learning in algebra, functions, coordinate geometry, and other advanced mathematical topics.

Advantages of NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

The NCERT Solutions for Class 10 Maths Chapter 2 Polynomials can be useful for students who want to understand the chapter thoroughly and practise NCERT textbook questions.

Step-by-Step Solutions

Each question is explained using a clear sequence of steps so students can understand how the final answer is obtained.

Better Conceptual Understanding

The solutions explain important concepts such as the zeroes of a polynomial, polynomial graphs, and the relationship between zeroes and coefficients.

Easy Revision

Students can use the solutions to quickly revise Class 10 Maths Chapter 2 Exercise 2.1 and Exercise 2.2 before examinations.

Helps Identify Mistakes

Comparing your method with the step-by-step solution can help you identify calculation, sign, and factorisation errors.

Useful for Self-Study

Students can use the Polynomials Class 10 NCERT Solutions to practise independently and clear doubts while studying at home.

Supports Exam Preparation

Regular practice of NCERT questions helps students become familiar with the concepts and question patterns included in the chapter.

NCERT Solutions for Class 10 Maths Chapter-Wise PDF Free Download

Download CBSE Class 10 Maths Notes PDF

Students can read each chapter online or use the chapter-wise PDF links below for offline revision.