NCERT Solutions Class 10 Maths Chapter 2 Polynomials

Class 10 Math Polynomials" is crucial as it covers advanced mathematical concepts. This chapter takes a closer look at polynomials, examining their types, properties, and operations, all of which are essential for understanding algebra and calculus in higher education. A solid understanding of polynomials enhances problem-solving skills and prepares students for complex equations and real-world applications.

The NCERT Solutions for Class 10 Maths polynomials offers comprehensive explanations and detailed, step-by-step solutions to textbook exercises. This resource simplifies complex topics such as polynomial identities, factorisation, and the remainder and factor theorems, making it easier for students to grasp challenging concepts. Our blog helps students excel in this chapter, improve their analytical abilities, and build confidence in mathematics. 

1.0NCERT Solutions Class 10 Maths Chapter 2 Polynomials: Free PDF

Downloading the NCERT Solutions for Class 10 Polynomials PDF is an excellent way to understand the key concepts outlined in the curriculum. This resource will enhance your understanding and provide valuable insights into the topic. Download the NCERT Solutions for Class 10 Maths Polynomials PDF from below. It gives you access to full explanations and step-by-step solutions to help you excel in your studies.

2.0NCERT Solutions Polynomials for Class 10 Maths Chapter 2: All Exercises

NCERT Solutions Class 10 Maths, Chapter 2: Polynomials, covers essential topics such as the geometrical meaning of zeros in a polynomial, the relationship between zeros and coefficients in a polynomial, and the division algorithm for polynomials. This provides a solid foundation for understanding more complex mathematical ideas. The NCERT Solution consists of different types of questions based on all these topics. 

3.0NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Overview

The NCERT Solutions for Class 10 Maths provides sufficient questions to find the zeroes of the given quadratic polynomials, and you will understand the relationship between the zeroes and the coefficients. This chapter's NCERT Solutions make the concepts easier to understand by providing in-depth justifications, real-world examples, and step-by-step solutions for the exercises. 

4.0Important Topics of Class 10 Maths Chapter 2 Polynomials  Solutions 

Chapter 2 of Class 10 Mathematics, "Polynomials," explores the basics of polynomials, including their definition, types, and important concepts like zeros, degrees, and factors. Key topics include the Remainder Theorem, Factor Theorem, and the relationship between the zeros and coefficients of a polynomial. Understanding how to factor polynomials and find their zeros is essential for solving algebraic equations. This chapter builds a foundation for advanced topics in algebra and geometry.

1. Definition of Polynomials 
2. Types of Polynomials   
3. Zero of a Polynomial  
4. Remainder Theorem 
5. Factor Theorem 
6. Factorization of Polynomials
7. Graph of Polynomials
8. Relationship between the Zeros and Coefficients of a Polynomial

5.0General Outline for Class 10 Maths Chapter 2 Polynomials

Introduction to Polynomials

Definition: A polynomial is an algebraic expression that consists of variables, constants, and exponents, combined using addition, subtraction, and multiplication (but not division by variables).

General form: $a{x}^n+b{x}^{n-1}+\cdots +c$, where $a,b,\dots ,c$ are constants and n is a non-negative integer.

Types of Polynomials:

• Monomial: A polynomial with only one term (e.g., 2x).
• Binomial: A polynomial with two terms (e.g., x + 2x + 2).
• Trinomial: A polynomial with three terms (e.g., x^2 + 3x + 2).

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable in the polynomial.
• Examples:
• • For $x^2+2x+1$, the degree is 2.
  • For $5x^4+3x^2+2x+1$, the degree is 4.

Zeros of a Polynomial

• Definition: The zeros (or roots) of a polynomial are the values of x for which the polynomial evaluates to zero.
• Finding Zeros:
• • Solve the polynomial equation f(x) = 0.
  • Example: For $x^2-5x+6=0$, the zeros are x = 2 and x = 3.
• Relationship between Polynomial and its Zeros: If x = r is a zero of the polynomial f(x), then $f(x)=(x-r)\times \text{(another factor)}$.

Remainder Theorem

• Statement: When a polynomial f(x) is divided by (x − a), the remainder is f(a).
• Example: If $f(x)=x^2-3x+2$ and divided by (x − 1), then the remainder is $f(1)=1^2–3(1)+2=0.$

Factorization of Polynomials

• Factorization: Expressing a polynomial as a product of its factors.
• Methods of Factorization:
• • By splitting the middle term: For quadratics like $a{x}^2+bx+c.$
  • Using the factor theorem: If x = r is a zero of the polynomial, then (x – r) is a factor.
• Example: Factorize $6x^2-5x+6$ by splitting the middle term: (x − 2)(x − 3).

Algebraic Identities

Standard Identities:

• $(a+b{)}^2=a^2+2ab+b^2$
• $(a-b{)}^2=a^2-2ab+b^2$
• $(a+b)(a-b)=a^2-b^2$

Application in Factorization: Using these identities to simplify and factor polynomials.

Graph of a Polynomial

Sketching the Graph: Understanding the shape of the graph of a polynomial based on the degree and the number of zeros.

• For even-degree polynomials, the graph generally has the same direction on both sides.
• For odd-degree polynomials, the graph typically has opposite directions on both sides.

6.0Sample of NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

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

Solutions:

(i) Graph of y=p(x) does not intersect the x-axis. Hence, polynomial p(x) has no zero.

(ii) Graph of y=p(x) intersects the x-axis at one and only one point. Hence, polynomial p(x) has one and only one real zero.

(iii) Graph of y=p(x) intersects the x-axis at 3 points. Hence, polynomial p(x) has 3 zeros.

(iv) Graph of y=p(x) intersects the x-axis at 2 points. Hence, polynomial p(x) has 2 zeros.

(v) Graph of y=p(x) intersects the x-axis at 4 points. Hence, polynomial p(x) has 4 zeros.

(vi) Graph of y=p(x) intersects the x-axis at 1 point and touches the x-axis at 2 points. Hence, p(x) has 3 zeros.

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

(i) x² - 2x - 8

x² - 2x - 8 = x² - 4x + 2x - 8 = x(x - 4) + 2(x - 4) = (x + 2)(x - 4)

Zeros are -2 and 4.

Sum of zeros = (-2) + (4) = 2 = -(-2)/1 = -(Coefficient of x) / (Coefficient of x²)

Product of zeros = (-2)(4) = -8 = -8/1 = (Constant term) / (Coefficient of x²)

(ii) 4s² - 4s + 1

4s² - 4s + 1 = (2s - 1)²

Zeros are 1/2, 1/2.

Sum of zeros = 1/2 + 1/2 = 1 = -(-4)/4 = -(Coefficient of s) / (Coefficient of s²)

Product of zeros = (1/2)(1/2) = 1/4 = (Constant term) / (Coefficient of s²)

(iii) 6x² - 3 - 7x

6x² - 7x - 3 = 6x² - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (2x - 3)(3x + 1)

Zeros are 3/2, -1/3.

Sum of zeros = 3/2 + (-1/3) = 7/6 = -(-7)/6 = -(Coefficient of x) / (Coefficient of x²)

Product of zeros = (3/2)(-1/3) = -1/2 = -3/6 = (Constant term) / (Coefficient of x²)

(iv) 4u² + 8u

4u² + 8u = 4u(u + 2)

Zeros are 0, -2.

Sum of zeros = 0 + (-2) = -2 = -8/4 = -(Coefficient of u) / (Coefficient of u²)

Product of zeros = 0(-2) = 0 = 0/4 = (Constant term) / (Coefficient of u²)

(v) t² - 15

t² - 15 = (t - √15)(t + √15)

Zeros are √15, -√15.

Sum of zeros = √15 + (-√15) = 0 = 0/1 = -(Coefficient of t) / (Coefficient of t²)

Product of zeros = (√15)(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t²)

(vi) 3x² - x - 4

3x² - x - 4 = 3x² - 4x + 3x - 4 = x(3x - 4) + 1(3x - 4) = (3x - 4)(x + 1)

Zeros are 4/3, -1.

Sum of zeros = 4/3 - 1 = 1/3 = -(-1)/3 = -(Coefficient of x) / (Coefficient of x²)

Product of zeros = (4/3)(-1) = -4/3 = (Constant term) / (Coefficient of x²)

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

(i) 1/4, -1

Required polynomial = x² - (sum of zeros)x + (product of zeros) = x² - (1/4)x - 1 = (1/4)(4x² - x - 4)

(ii) √2, 1/3

Required polynomial = x² - (sum of zeros)x + (product of zeros) = x² - √2x + 1/3 = (1/3)(3x² - 3√2x + 1)

(iii) 0, √5

Required polynomial = x² - (sum of zeros)x + (product of zeros) = x² - 0x + √5 = x² + √5

(iv) 1, 1

Required polynomial = x² - (sum of zeros)x + (product of zeros) = x² - 1x + 1 = x² - x + 1

(v) -1/4, 1/4

Required polynomial = x² - (sum of zeros)x + (product of zeros) = x² - (-1/4)x + 1/4 = x² + (1/4)x + 1/4 = (1/4)(4x² + x + 1)

(vi) 4, 1

Required polynomial = x² - (sum of zeros)x + (product of zeros) = x² - 4x + 1

7.0Benefits of Class 10 Maths Chapter 2 Polynomials

Polynomials play a crucial role in Class 10 Mathematics. Here’s why they’re important:

1. Foundation for Advanced Topics: Polynomials prepare you for higher-level concepts in algebra, calculus, and quadratic equations.
2. Improved Problem-Solving: They enhance your ability to solve algebraic equations and simplify complex problems.
3. Real-World Applications: Polynomials are used to model situations like population growth, physics problems, and finance.
4. Stronger Algebra Skills: Mastering polynomials improves your skills in factoring, simplifying, and solving equations.
5. Better Understanding of Graphs: Polynomials help in understanding the behavior of functions and interpreting their graphs.
6. Boosts Logical Thinking: Working with polynomials strengthens your logical reasoning and analytical skills.
7. Competitive Exam Preparation: They frequently appear in competitive exams, making them essential for success.

8.0Advantages of NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

• Comprehensive Step-by-Step Guidance: Each exercise is carefully explained with detailed, step-by-step solutions, helping students navigate through problems confidently.
• Enhanced Understanding of Concepts: These solutions offer clear and detailed explanations, making it easier for students to grasp the core principles of polynomials.
• Solid Mathematical Foundation: These NCERT Solutions can strengthen students' understanding of polynomials, which is essential for advancing in mathematics.
• Focused Exam Preparation: Practicing with these solutions helps students familiarize themselves with potential exam questions, improving their readiness and boosting their chances of success.