CBSE Class 10 Maths Notes – Chapter 2 Polynomials (NCERT Solutions + Free PDF)

CBSE Class 10 Maths Notes – Chapter 2 Introduction to Polynomials

In This post CBSE Class 10 Notes: Polynomials Chapter 2 (NCERT Solutions)
Description: Get free CBSE Class 10 Maths Notes for Chapter 2 Introduction to Polynomials. NCERT-based notes with formulas, solved examples, graphs & important board questions.

Welcome to the most comprehensive CBSE Class 10 Notes for Mathematics Chapter 2. These CBSE Class 10 Maths Notes are meticulously crafted for board exam preparation, strictly following the NCERT curriculum. Whether you're aiming for full marks in your school exams or building a strong foundation for competitive tests, this guide covers everything from basic definitions to advanced HOTS questions.

1. Introduction

Polynomials form the backbone of algebraic mathematics and are crucial for CBSE Class 10 board examinations. This chapter introduces you to expressions involving variables and coefficients, helping you understand how to analyze, factorize, and graph polynomial functions.

Why are Polynomials Important?

• High Weightage: Carries 4-6 marks in CBSE board exams
• Foundation for Advanced Math: Essential for Class 11-12 Algebra and Calculus
• Real-world Applications: Used in physics, engineering, and economics for modeling relationships

These CBSE Class 10 Maths Notes are 100% aligned with NCERT Chapter 2 and cover all concepts prescribed by the CBSE syllabus.

2. Chapter Overview

Aspect	Details
Topics Covered	Definition, Types, Zeroes, Coefficient Relations, Graphs
Marks Weightage	4-6 marks (Board Exam)
Question Types	MCQs, Short Answer (2-3 marks), Long Answer (4-5 marks)
Difficulty Level	Easy to Moderate
NCERT Exercises	2.1, 2.2, 2.3, 2.4 + Optional Exercise

CBSE Class 10 Notes Tip: Focus heavily on the relationship between zeroes and coefficients—this topic appears every year without fail!

3. Key Concepts (Exam-Oriented)

Definition of a Polynomial

A polynomial in one variable x is an algebraic expression of the form:

p(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

Where:

• a_n, a_n-1, ..., a₀ are real numbers (coefficients)
• n is a non-negative integer (degree)
• x is the variable

Important: Powers of x must be whole numbers (0, 1, 2, 3...). Expressions like x^-1 or x^1/2 are NOT polynomials.

Terms and Coefficients

• Term: Each part of the polynomial separated by + or - signs
• Coefficient: The numerical factor of each term

Example: In 3x² - 5x + 7

• Terms: 3x², -5x, 7
• Coefficients: 3, -5, 7

Degree of a Polynomial

The highest power of the variable in the polynomial.

Polynomial	Degree
5	0
2x + 3	1
x2 - 4x + 4	2
x3 - 2x2 + x - 1	3

Types of Polynomials

Type	Degree	General Form	Example
Constant	0	p(x) = c	p(x) = 5
Linear	1	p(x) = ax + b	p(x) = 2x + 3
Quadratic	2	p(x) = ax2 + bx + c	p(x) = x2 - 5x + 6
Cubic	3	p(x) = ax3 + bx2 + cx + d	p(x) = x3 - 1

Zeroes of a Polynomial

A real number k is a zero of polynomial p(x) if p(k) = 0.

Geometric Meaning: Zeroes are the x-coordinates where the graph of y = p(x) intersects the x-axis.

Relationship Between Zeroes and Coefficients

For Linear Polynomial ax + b:

Zero = -b/a

For Quadratic Polynomial ax² + bx + c:

• Sum of zeroes: α + β = -b/a
• Product of zeroes: αβ = c/a

For Cubic Polynomial ax³ + bx² + cx + d:

• Sum of zeroes: α + β + γ = -b/a
• Sum of products taken two at a time: αβ + βγ + γα = c/a
• Product of zeroes: αβγ = -d/a

4. Important Formulas

Must-Remember Formulas for CBSE Class 10 Maths Notes

Polynomial Type	Formula	When to Use
Linear	Zero = -b/a	Finding the root of linear equations
Quadratic	Sum: α + β = -b/a	When you know coefficients and need sum of roots
Quadratic	Product: αβ = c/a	When you know coefficients and need product of roots
Quadratic Formation	x2 - (α+β)x + αβ = 0	To form equation when zeroes are given

Memory Trick:

• Sum = -(coefficient of x)/(coefficient of x²) → "Minus b by a"
• Product = +(constant term)/(coefficient of x²) → "c by a"

5. Solved Examples (CBSE Pattern)

1 Mark Example 1: Find the zero of the polynomial p(x) = 3x - 6.

Solution:
Set p(x) = 0
3x - 6 = 0
3x = 6
x = 2

Answer: The zero is 2.

3 Marks Example 2: Find the zeroes of the quadratic polynomial x² - 5x + 6 and verify the relationship between zeroes and coefficients.

Solution:

Step 1: Factorize the polynomial
x² - 5x + 6 = x² - 3x - 2x + 6
= x(x - 3) - 2(x - 3)
= (x - 3)(x - 2)

Step 2: Find zeroes
(x - 3)(x - 2) = 0
Zeroes are: α = 3 and β = 2

Step 3: Verify relationships
• Sum of zeroes: α + β = 3 + 2 = 5
• Formula check: -b/a = -(-5)/1 = 5 ✓

• Product of zeroes: αβ = 3 × 2 = 6
• Formula check: c/a = 6/1 = 6 ✓

Hence verified.

4 Marks Example 3 (Board-Level): If α and β are the zeroes of the polynomial 6x² - 7x + 2, find a quadratic polynomial whose zeroes are 1/α and 1/β.

Solution:

Step 1: Find sum and product of given zeroes
For 6x² - 7x + 2:
• α + β = -b/a = 7/6
• αβ = c/a = 2/6 = 1/3

Step 2: Find sum of new zeroes
1/α + 1/β = (α + β)/(αβ) = (7/6)/(1/3) = (7/6) × 3 = 7/2

Step 3: Find product of new zeroes
(1/α) × (1/β) = 1/(αβ) = 1/(1/3) = 3

Step 4: Form new polynomial
x² - (sum)x + (product) = 0
x² - (7/2)x + 3 = 0

Multiply by 2 to clear fractions:
2x² - 7x + 6 = 0

Answer: The required polynomial is 2x² - 7x + 6 (or any non-zero multiple of it).

6. Smart Tricks Section

Quick Degree Identification Trick

Look at the highest power of x:

• No x → Degree 0 (Constant)
• x¹ → Degree 1 (Linear)
• x² → Degree 2 (Quadratic)
• x³ → Degree 3 (Cubic)

Shortcut: Ignore coefficients, just check the exponent!

Fast Zero Checking Trick

To verify if k is a zero of p(x), simply calculate p(k). If result is 0, it's a zero!

Example: Is x = 2 a zero of x³ - 8?
p(2) = 2³ - 8 = 8 - 8 = 0
Yes! (Because x³ - 8 = (x-2)(x²+2x+4))

Graph Interpretation Shortcut

Graph Behavior	Number of Zeroes
Straight line crossing x-axis once	1 zero
Parabola touching x-axis at one point	1 zero (repeated)
Parabola cutting x-axis at two points	2 zeroes
Parabola not touching x-axis	0 zeroes (real)

MCQ Elimination Trick

If asked to find zeroes and options are given:

1. Substitute each option into the polynomial
2. The one giving zero is your answer
3. Saves time in board exams!

7. Visual Learning

Graph of Linear Polynomial

Key Points:

• Always a straight line
• Cuts x-axis at exactly one point
• General form: y = ax + b
• One real zero guaranteed

Graph of Quadratic Polynomial (Parabola)

Three Cases for Zeroes:

1. Two Distinct Zeroes: Parabola cuts x-axis at two different points (b² - 4ac > 0)
2. One Repeated Zero: Parabola touches x-axis at one point (b² - 4ac = 0)
3. No Real Zeroes: Parabola doesn't touch x-axis (b² - 4ac < 0)

Comparison: Linear, Quadratic, and Cubic

8. Most Important Board Questions

★ 1 Mark Questions

Q1. Find the degree of the polynomial 5x³ - 3x² + 2x - 7.
Ans: 3

Q2. If one zero of x² - 5x + k is 2, find the value of k.
Ans: 4 - 10 + k = 0 ⇒ k = 6

Q3. How many zeroes does a linear polynomial have?
Ans: Exactly one zero.

★★ 2-3 Mark Questions

Q4. Find a quadratic polynomial whose sum and product of zeroes are -3 and 2 respectively.
Solution: x² - (-3)x + 2 = x² + 3x + 2

Q5. If α and β are zeroes of 2x² - 3x + 1, find α² + β².
Solution:
α + β = 3/2
αβ = 1/2
α² + β² = (α+β)² - 2αβ = 9/4 - 1 = 5/4

★★★ 4-5 Mark Proof-Based Questions

Q6. Repeated every year
If α and β are zeroes of x² - 6x + a and 3α + 2β = 20, find the value of a.

Solution:
From polynomial: α + β = 6 and αβ = a

Given: 3α + 2β = 20

From first equation: β = 6 - α

Substitute:
3α + 2(6 - α) = 20
3α + 12 - 2α = 20
α = 8

Then β = 6 - 8 = -2

Therefore: a = αβ = 8 × (-2) = -16

Q7. High Probability
Find all zeroes of 2x⁴ - 3x³ - 3x² + 6x - 2 if two of its zeroes are √2 and -√2.

Solution:
Since √2 and -√2 are zeroes, (x - √2)(x + √2) = x² - 2 is a factor.

Divide the polynomial by x² - 2 to get 2x² - 3x + 1.

Factorize: 2x² - 3x + 1 = (2x - 1)(x - 1)

Other zeroes: x = 1/2 and x = 1

All zeroes: √2, -√2, 1/2, 1

9. Common Mistakes Students Make

Sign Errors

Mistake: Writing sum of zeroes as b/a instead of -b/a

Correction: Always remember the minus sign for sum!

Degree Confusion

Mistake: Thinking 3x² + 5x^-1 + 2 is degree 2 polynomial

Correction: Negative powers make it not a polynomial at all!

Skipping Steps in Verification

Mistake: Directly writing "Hence verified" without showing calculations

Correction: Board examiners give marks for step-by-step verification. Always show:

1. Calculate from zeroes
2. Calculate from formula
3. Compare both results

Writing Formula Incorrectly

Mistake: Confusing product formula with sum formula

Memory Aid:
• Sum → Starts with Sign (minus)
• Product → Positive (no sign change)

10. Practice Section

MCQs

Q1. The zero of p(x) = 2x - 5 is:
(a) 5/2 (b) -5/2 (c) 2/5 (d) -2/5

Q2. A quadratic polynomial can have:
(a) At most 2 zeroes (b) Exactly 2 zeroes (c) At least 2 zeroes (d) 0 or 1 or 2 zeroes

Q3. If sum of zeroes is 5 and product is 6, the polynomial is:
(a) x² + 5x + 6 (b) x² - 5x + 6 (c) x² - 5x - 6 (d) x² + 5x - 6

Answers: 1-(a), 2-(d), 3-(b)

Assertion-Reason Questions

Q4.
Assertion (A): The polynomial x² + 1 has no real zeroes.
Reason (R): A quadratic polynomial always has two real zeroes.

(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

Answer: (c) A is true but R is false (A quadratic can have 0, 1, or 2 real zeroes)

Case Study Question

Q5. A gardener wants to plant trees in a rectangular park. The area of the park is represented by the polynomial x² - 5x + 6 and the length is (x - 2) meters.

(i) Find the breadth of the park.
(ii) If x = 4, find the actual dimensions.
(iii) Find the zeroes of the area polynomial and interpret their meaning.

Solution:

(i) Area = Length × Breadth
x² - 5x + 6 = (x - 2)(x - 3)
So breadth = (x - 3) meters

(ii) When x = 4:
• Length = 4 - 2 = 2 meters
• Breadth = 4 - 3 = 1 meter

(iii) Zeroes are x = 2 and x = 3
• At x = 2: Breadth becomes 0 (degenerate rectangle)
• At x = 3: Length becomes 0 (degenerate rectangle)

HOTS Question

Q6. If α and β are zeroes of x² - px + q, find the value of:
(α²/β²) + (β²/α²)

Solution:
Given: α + β = p and αβ = q

(α²/β²) + (β²/α²) = (α⁴ + β⁴)/(α²β²)

= [(α² + β²)² - 2α²β²]/(αβ)²

= [((α+β)² - 2αβ)² - 2(αβ)²]/(αβ)²

= [(p² - 2q)² - 2q²]/q²

= [p⁴ - 4p²q + 4q² - 2q²]/q²

= (p⁴ - 4p²q + 2q²)/q²

FAQ Section

Q1. Is Polynomials important for CBSE Class 10 board exam?

Yes! Polynomials is one of the most scoring chapters in CBSE Class 10 Maths Notes. It carries 4-6 marks and concepts are repeatedly asked every year. The relationship between zeroes and coefficients is a favorite topic of CBSE.

Q2. How to score full marks in Chapter 2 Introduction to Polynomials?

To score full marks:

1. Memorize all formulas perfectly (especially the sign in sum of zeroes)
2. Practice NCERT examples and exercises thoroughly
3. Learn to form polynomials when zeroes are given
4. Master the graphical interpretation of zeroes
5. Solve previous 5 years' board questions

Q3. Are these CBSE Class 10 Maths Notes based on NCERT?

Absolutely! These CBSE Class 10 Notes are 100% aligned with NCERT Chapter 2 Introduction to Polynomials. Every concept, formula, and solved example follows the NCERT pattern preferred by CBSE board examiners.

Q4. What is the weightage of Chapter 2 in CBSE Class 10 board exam?

Chapter 2 Polynomials typically carries 4-6 marks in the CBSE Class 10 Mathematics board exam. Questions appear as 1-mark MCQs, 2-mark short answers, and 4-mark long answer types.

Q5. Can I download these Class 10 Maths Notes PDF?

These CBSE Class 10 Maths Notes are optimized for both web viewing and PDF conversion. You can save this page or print to PDF for offline revision during your board exam preparation.

Conclusion

Mastering CBSE Class 10 Notes for Chapter 2 Introduction to Polynomials requires consistent practice and conceptual clarity. Remember:

• Formulas are your foundation—memorize them perfectly
• NCERT is your bible—solve every example and exercise
• Previous year papers reveal the pattern—practice them religiously

Revision Strategy:

1. Review these CBSE Class 10 Maths Notes weekly
2. Create formula flashcards for quick revision
3. Teach the concepts to a friend (best way to learn!)
4. Time yourself while solving problems

Suggested Next Topics: After mastering Polynomials, continue with:

• CBSE Class 10 Notes Chapter 3: Pair of Linear Equations in Two Variables
• CBSE Class 10 Maths Notes Chapter 4: Quadratic Equations
• CBSE Class 10 Notes Chapter 8: Introduction to Trigonometry

Best of luck for your CBSE Class 10 Board Exams!

These CBSE Class 10 Maths Notes are prepared by expert mathematics faculty with 15+ years of teaching experience. For any doubts or queries regarding Introduction to Polynomials Class 10, feel free to ask in the comments section.

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