How to Solve RD Sharma Exercise 2.1?
To solve RD Sharma Class 10 Exercise 2.1 (Q1–Q6) on Polynomials, find the zeros of each quadratic polynomial by factorization or the quadratic formula. Verify the relationship between zeros and coefficients using: Sum of zeros = –b/a, Product of zeros = c/a. Watch our video explanation for all solutions: YouTube Video.
Table of Contents
Polynomials are a cornerstone of the Class 10 Mathematics syllabus, vital for CBSE board exam success. RD Sharma’s textbook is a trusted resource, offering rigorous exercises to build conceptual clarity. Exercise 2.1 in Chapter 2 focuses on finding zeros of quadratic polynomials and verifying their relationship with coefficients. This SEO-optimized, plagiarism-free guide provides detailed solutions for questions 1 to 6 of RD Sharma Class 10 Exercise 2.1, with step-by-step explanations and a comprehensive video tutorial. Perfect for students and educators, this resource ensures a thorough understanding of polynomials.
Understanding Polynomials and Exercise 2.1
A polynomial is an algebraic expression with variables and coefficients, combined using addition, subtraction, and multiplication. In Class 10, quadratic polynomials of the form ax² + bx + c (where a ≠ 0) are emphasized. The zeros of a polynomial are values of the variable that make the polynomial equal to zero. Exercise 2.1 requires students to find these zeros and verify their sum and product using:
- Sum of zeros = –b/a
- Product of zeros = c/a
Questions 1 to 6 involve applying these concepts to specific quadratic polynomials. Our solutions below are designed for clarity and exam readiness.
Detailed Solutions for Exercise 2.1 (Q1–Q6)
Question 1: Find the zeros of the quadratic polynomial f(x) = x² – 2x – 8 and verify the relationship between the zeros and their coefficients.
Step-by-Step Solution:
- Find the zeros: Set f(x) = x² – 2x – 8 = 0.
- Factorize: x² – 2x – 8 = (x – 4)(x + 2).
- Solve: (x – 4)(x + 2) = 0 gives x = 4 or x = –2. Zeros are 4 and –2.
- Verify: For ax² + bx + c, a = 1, b = –2, c = –8.
- Sum of zeros = 4 + (–2) = 2. Compare with –b/a = –(–2)/1 = 2. Verified.
- Product of zeros = 4 × (–2) = –8. Compare with c/a = –8/1 = –8. Verified.
Question 2: Find the zeros of the quadratic polynomial f(x) = 4s² – 4s + 1 and verify the relationship between the zeros and their coefficients.
Step-by-Step Solution:
- Find the zeros: Set f(s) = 4s² – 4s + 1 = 0.
- Factorize: 4s² – 4s + 1 = (2s – 1)(2s – 1) = (2s – 1)².
- Solve: (2s – 1)² = 0 gives s = 1/2. Zeros are 1/2 and 1/2 (repeated).
- Verify: a = 4, b = –4, c = 1.
- Sum of zeros = 1/2 + 1/2 = 1. Compare with –b/a = –(–4)/4 = 1. Verified.
- Product of zeros = 1/2 × 1/2 = 1/4. Compare with c/a = 1/4. Verified.
Question 3: Find the zeros of the quadratic polynomial f(x) = 6x² – 3 – 7x and verify the relationship between the zeros and their coefficients.
Step-by-Step Solution:
- Find the zeros: Rewrite as 6x² – 7x – 3 = 0.
- Factorize: 6x² – 7x – 3 = (3x + 1)(2x – 3).
- Solve: (3x + 1)(2x – 3) = 0 gives x = –1/3 or x = 3/2. Zeros are –1/3 and 3/2.
- Verify: a = 6, b = –7, c = –3.
- Sum of zeros = –1/3 + 3/2 = 7/6. Compare with –b/a = –(–7)/6 = 7/6. Verified.
- Product of zeros = (–1/3) × (3/2) = –1/2. Compare with c/a = –3/6 = –1/2. Verified.
Question 4: Find the zeros of the quadratic polynomial f(x) = x² – 15 and verify the relationship between the zeros and their coefficients.
Step-by-Step Solution:
- Find the zeros: Set f(x) = x² – 15 = 0.
- Solve: x² = 15, so x = ±√15. Zeros are √15 and –√15.
- Verify: a = 1, b = 0, c = –15.
- Sum of zeros = √15 + (–√15) = 0. Compare with –b/a = –0/1 = 0. Verified.
- Product of zeros = √15 × (–√15) = –15. Compare with c/a = –15/1 = –15. Verified.
Question 5: Find the zeros of the quadratic polynomial f(x) = x² – 3x – 2 and verify the relationship between the zeros and their coefficients.
Step-by-Step Solution:
- Find the zeros: Set f(x) = x² – 3x – 2 = 0.
- Use quadratic formula: x = [–b ± √(b² – 4ac)]/(2a), where a = 1, b = –3, c = –2.
- Calculate: x = [3 ± √(9 + 8)]/2 = [3 ± √17]/2. Zeros are (3 + √17)/2 and (3 – √17)/2.
- Verify:
- Sum of zeros = [(3 + √17)/2] + [(3 – √17)/2] = 3. Compare with –b/a = –(–3)/1 = 3. Verified.
- Product of zeros = [(3 + √17)/2] × [(3 – √17)/2] = –2. Compare with c/a = –2/1 = –2. Verified.
Question 6: Find the zeros of the quadratic polynomial f(x) = 9x² – 18x + 8 and verify the relationship between the zeros and their coefficients.
Step-by-Step Solution:
- Find the zeros: Set f(x) = 9x² – 18x + 8 = 0.
- Factorize: 9x² – 18x + 8 = (3x – 2)(3x – 4).
- Solve: (3x – 2)(3x – 4) = 0 gives x = 2/3 or x = 4/3. Zeros are 2/3 and 4/3.
- Verify: a = 9, b = –18, c = 8.
- Sum of zeros = 2/3 + 4/3 = 2. Compare with –b/a = –(–18)/9 = 2. Verified.
- Product of zeros = (2/3) × (4/3) = 8/9. Compare with c/a = 8/9. Verified.
Video Explanation: Watch our comprehensive video covering all six questions to visualize factorization, quadratic formula application, and verification steps.
How to Solve Polynomial Questions
Follow these steps to tackle polynomial questions like those in Exercise 2.1:
- Identify the Polynomial: Ensure it’s in the form ax² + bx + c.
- Find Zeros: Use factorization (split the middle term) or the quadratic formula x = [–b ± √(b² – 4ac)]/(2a).
- Verify Relationships: Calculate the sum and product of zeros and compare with –b/a and c/a.
- Practice Regularly: Solve similar problems to build speed and accuracy.
- Use Visual Aids: Refer to our video for graphical insights into polynomial zeros.
Why Choose Our RD Sharma Solutions?
- Clear Explanations: Step-by-step solutions in simple language.
- Video Support: A single video explains all six questions comprehensively.
- CBSE-Aligned: Optimized for Class 10 board exam preparation.
Frequently Asked Questions
What is the purpose of RD Sharma Exercise 2.1?
Exercise 2.1 focuses on finding zeros of quadratic polynomials and verifying their relationship with coefficients, building a strong foundation in polynomial concepts for Class 10 students.
How do I find the zeros of a quadratic polynomial?
You can find zeros by factorizing the polynomial (e.g., splitting the middle term) or using the quadratic formula x = [–b ± √(b² – 4ac)]/(2a).
Why verify the sum and product of zeros?
Verification confirms the zeros are correct by ensuring their sum equals –b/a and their product equals c/a, reinforcing the polynomial’s properties.
Is the video explanation necessary?
While the written solutions are comprehensive, the video provides visual clarity, especially for complex factorizations or quadratic formula applications.
Conclusion
RD Sharma’s Exercise 2.1 on Polynomials is essential for mastering quadratic polynomials in Class 10. Our detailed, SEO-friendly solutions for questions 1 to 6, paired with a single video explanation, offer a complete learning package. Practice these problems, use our step-by-step guide, and leverage the video to excel in your CBSE exams. Explore our website for more Class 10 Mathematics resources and stay ahead in your studies!
Class 10 RD Sharma Exercise 2.1 (Q1–Q6) Solutions | Polynomials
Q: What is the focus of RD Sharma Class 10 Exercise 2.1?
A: It focuses on finding the zeros of quadratic polynomials and verifying the relationship between the zeros and their coefficients.
Table of Contents
- Introduction to Polynomials
- What Are Zeros of Polynomials?
- Relation Between Zeros and Coefficients
- RD Sharma Exercise 2.1 Q1–Q6 Solutions
- Video Explanation
- FAQs on Polynomials
Introduction to Polynomials
Polynomials are algebraic expressions involving constants, variables, and powers of variables. They are fundamental in Class 10 maths and form the base of algebraic operations.
What Are Zeros of Polynomials?
The zeros (or roots) of a polynomial are the values of x for which the polynomial becomes zero. For quadratic polynomials, they can be found using factorization or the quadratic formula.
Relation Between Zeros and Coefficients
In a polynomial of the form ax² + bx + c:
- Sum of zeros = -b/a
- Product of zeros = c/a
This relationship helps verify whether the obtained zeros are correct.
RD Sharma Exercise 2.1 Q1–Q6 Solutions
Below are the first few questions from Exercise 2.1 of Chapter 2:
Q1. Find the zeros of x² - 2x - 8 and verify the relation between zeros and coefficients.
Solution:
x² - 2x - 8 = (x - 4)(x + 2) ⇒ Zeros: 4, -2
Sum = 4 + (-2) = 2 = -(-2)/1
Product = 4 × (-2) = -8 = -8/1
Hence, verified.
Q2. Find the zeros of x² + x - 6 and verify the relation.
Solution:
x² + x - 6 = (x + 3)(x - 2) ⇒ Zeros: -3, 2
Sum = -1 = -1/1, Product = -6 = -6/1 ⇒ Verified.
Q3. x² + 5x + 6
(x + 2)(x + 3) ⇒ Zeros: -2, -3. Sum = -5, Product = 6 ⇒ Verified.
Q4. x² - 7x + 10
(x - 5)(x - 2) ⇒ Zeros: 5, 2. Sum = 7, Product = 10 ⇒ Verified.
Q5. x² - x - 6
(x - 3)(x + 2) ⇒ Zeros: 3, -2. Sum = 1, Product = -6 ⇒ Verified.
Q6. x² + 4x + 4
(x + 2)² ⇒ Zero repeated: -2. Sum = -4, Product = 4 ⇒ Verified.
Video Explanation
FAQs on Polynomials
What is a polynomial?
A polynomial is an algebraic expression made up of variables and constants using operations like addition, subtraction, and multiplication.
How to find the zeros of a quadratic polynomial?
Use factorization, completing the square, or the quadratic formula to find the zeros of a quadratic polynomial.
What is the formula for sum and product of zeros?
Sum of zeros = -b/a and Product of zeros = c/a for a quadratic polynomial ax² + bx + c.
Is RD Sharma enough for Class 10 board exam?
Yes, RD Sharma is excellent for concept clarity and practice. Combine it with NCERT for best results.
Can we skip quadratic formula if we know factorization?
Yes, but for difficult polynomials, the quadratic formula is a reliable alternative when factorization isn’t obvious.
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