Exercise 6.3 Class 10 Solution

Exercise 6.3 Class 10 Solution: A practical guide to Understanding Polynomials

This article provides a full breakdown to solving Exercise 6.Day to day, understanding polynomials is crucial for further studies in mathematics and related fields. Consider this: we'll cover each problem in detail, explaining the concepts and methods involved. This guide is designed to be accessible to all students, regardless of their prior knowledge, and will help build a solid foundation in this essential topic. We will break down the intricacies of each question, providing step-by-step solutions and explanations to ensure a thorough understanding. Day to day, 3 from Class 10 mathematics textbooks, focusing on polynomials. This detailed approach aims to not only provide answers but also equip students with the tools to tackle similar problems independently.

Introduction to Polynomials

Before diving into the solutions, let's refresh our understanding of polynomials. ) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Practically speaking, for example, 3x² + 2x - 5 is a polynomial of degree 2 (a quadratic polynomial), while 5x⁴ - 2x³ + x - 7 is a polynomial of degree 4. Exercise 6.The degree of a polynomial is the highest power of the variable present in the expression. A polynomial is an algebraic expression consisting of variables (usually denoted by x, y, etc.3 typically involves various operations on polynomials, such as finding the zeros, dividing polynomials, and verifying relationships between zeros and coefficients.

Exercise 6.3: Problem Breakdown and Solutions

Since I do not have access to a specific textbook's Exercise 6.3, Class 10, I will provide examples of common problems found in this exercise and their solutions. Even so, the specific problems in your textbook might vary slightly, but the principles and methods applied will be the same. Remember to always refer to your specific textbook for the exact questions.

Example Problem 1: Finding the Zeros of a Polynomial

Find the zeros of the polynomial p(x) = x² - 5x + 6.

Solution:

To find the zeros of a polynomial, we need to solve the equation p(x) = 0. In this case, we have:

x² - 5x + 6 = 0

This is a quadratic equation. We can solve it by factoring:

(x - 2)(x - 3) = 0

This equation is satisfied if either (x - 2) = 0 or (x - 3) = 0. That's why, the zeros of the polynomial are x = 2 and x = 3.

Example Problem 2: Division of Polynomials

Divide the polynomial 3x³ + x² - 20x + 12 by (x+3) using polynomial long division.

Solution:

Polynomial long division is a systematic method for dividing polynomials. Here's how to perform the division:

          3x² - 8x + 4
x + 3 | 3x³ + x² - 20x + 12
         - (3x³ + 9x²)
         ----------------
                   -8x² - 20x
                 - (-8x² - 24x)
                 ----------------
                           4x + 12
                         - (4x + 12)
                         ----------------
                                  0

The quotient is 3x² - 8x + 4 and the remainder is 0. This means (x+3) is a factor of 3x³ + x² - 20x + 12.

Example Problem 3: Relationship Between Zeros and Coefficients

If α and β are the zeros of the quadratic polynomial p(x) = 2x² - 5x + 7, find the values of:

a) α + β b) αβ

Solution:

For a quadratic polynomial of the form ax² + bx + c, the sum of the zeros (α + β) is given by -b/a, and the product of the zeros (αβ) is given by c/a. In this case, a = 2, b = -5, and c = 7. Therefore:

a) α + β = -(-5)/2 = 5/2 b) αβ = 7/2

Example Problem 4: Forming a Quadratic Polynomial

Form a quadratic polynomial whose zeros are 3 and -2.

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Solution:

If α and β are the zeros of a quadratic polynomial, the polynomial can be expressed as:

p(x) = (x - α)(x - β)

In this case, α = 3 and β = -2. Because of this, the quadratic polynomial is:

p(x) = (x - 3)(x + 2) = x² - x - 6

Example Problem 5: Remainder Theorem

Find the remainder when the polynomial p(x) = x³ - 3x² + 4x - 5 is divided by (x - 2).

Solution:

According to the Remainder Theorem, when a polynomial p(x) is divided by (x - a), the remainder is p(a). In this case, a = 2. Which means, the remainder is:

p(2) = (2)³ - 3(2)² + 4(2) - 5 = 8 - 12 + 8 - 5 = -1

Explanation of Underlying Mathematical Concepts

1. Factor Theorem: The Factor Theorem states that (x - a) is a factor of a polynomial p(x) if and only if p(a) = 0. This theorem is closely related to the Remainder Theorem and is often used in finding the zeros of a polynomial.

2. Remainder Theorem: As explained above, this theorem provides a shortcut for finding the remainder when a polynomial is divided by a linear divisor.

3. Polynomial Long Division: This is a crucial algebraic technique for dividing polynomials of higher degrees. It's essential for simplifying expressions and factoring polynomials.

4. Zeros of a Polynomial: The zeros of a polynomial are the values of x for which p(x) = 0. Finding the zeros is a fundamental problem in polynomial algebra. These zeros are also often referred to as roots.

5. Relationship between Zeros and Coefficients: There are established relationships between the zeros and the coefficients of a polynomial, particularly for quadratic and cubic polynomials, as demonstrated in Example Problem 3.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a zero and a root of a polynomial?

A1: The terms "zero" and "root" are often used interchangeably to represent the values of x for which p(x) = 0.

Q2: How do I choose the correct method for solving a polynomial problem?

A2: The choice of method depends on the specific problem. If you need to find the zeros, factoring or using the quadratic formula might be appropriate. In practice, for division, polynomial long division is typically used. Understanding the relationships between zeros and coefficients can simplify certain problems.

Q3: What if I get a non-zero remainder after polynomial long division?

A3: A non-zero remainder indicates that the divisor is not a factor of the polynomial. The result of the division is expressed as Quotient + Remainder/Divisor.

Q4: Are there other ways to find the zeros of a polynomial besides factoring?

A4: Yes, for quadratic equations, the quadratic formula can be used. For higher-degree polynomials, numerical methods or graphical techniques may be necessary.

Conclusion

This full breakdown provides a detailed walkthrough of common problems encountered in Exercise 6.Now, mastering these concepts is fundamental to success in higher-level mathematics. By carefully reviewing the examples and explanations provided, and by engaging in further practice problems, you can build a strong foundation in polynomial algebra and confidently tackle more complex problems in the future. Remember that consistent practice and a thorough understanding of the underlying mathematical principles are key to solving polynomial problems effectively. Practically speaking, 3 of Class 10 mathematics, focusing on polynomials. Remember to always refer to your textbook and consult with your teacher if you encounter any difficulties.