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Class 9 → Mathematics → Chapter 2: Polynomials
I. Chapter Summary
The chapter Polynomials introduces algebraic expressions consisting of variables and constants combined using mathematical operations like addition, subtraction, multiplication, and non-negative integer exponents. Students learn to classify polynomials based on degree and number of terms, understand the concept of zeroes, and explore relationships between zeroes and coefficients. The chapter also explains remainder and factor theorems, which are useful in solving polynomial equations.
II. Key Concepts Covered
1. Polynomial
An algebraic expression of the form:
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
2. Types of Polynomials
- Based on terms:
- Monomial (1 term)
- Binomial (2 terms)
- Trinomial (3 terms)
- Based on degree:
- Linear (degree 1)
- Quadratic (degree 2)
- Cubic (degree 3)
3. Degree of a Polynomial
The highest power of the variable.
4. Zeroes of a Polynomial
Values of x for which polynomial becomes zero.
5. Graph of Polynomials
- Linear → straight line
- Quadratic → parabola
6. Remainder Theorem
Remainder of p(x) ÷ (x – a) = p(a)
7. Factor Theorem
If p(a) = 0 → (x – a) is a factor
III. Important Questions
(A) MCQs (1 Mark)
- The degree of polynomial 5x³ + 2x² + x is:
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) - Number of zeroes of a linear polynomial is:
(a) 0 (b) 1 (c) 2 (d) infinite
Answer: (b) - If p(2) = 0, then (x – 2) is:
(a) Not a factor (b) Factor (c) Zero (d) None
Answer: (b) - Graph of a quadratic polynomial is:
(a) Line (b) Circle (c) Parabola (d) Hyperbola
Answer: (c)
(B) Short Answer Questions (2/3 Marks)
- Find the zeroes of polynomial x² – 5x + 6. (CBSE 2020)
- Verify remainder theorem for p(x) = x³ – 3x² + 2x and divisor (x – 1).
- Find whether (x + 2) is a factor of x³ + 8.
- Determine degree and type of polynomial 7x⁴ – 3x + 5.
(C) Long Answer Questions (5 Marks)
- Find zeroes of quadratic polynomial and verify relationship between zeroes and coefficients. (CBSE 2019)
- Use factor theorem to factorize x³ – 3x² – x + 3.
- Draw graph of linear polynomial and find zero graphically.
- Prove remainder theorem with suitable example.
(D) HOTS Questions
- If one zero of a quadratic polynomial is reciprocal of the other, what can you say about coefficients?
- A polynomial leaves remainder 2 when divided by (x – 1). What is p(1)?
IV. Key Formulas / Concepts
- Polynomial form:
p(x) = aₙxⁿ + … + a₀ - Remainder Theorem:
p(a) = remainder - Factor Theorem:
p(a) = 0 ⇒ (x – a) is factor - Relationship (Quadratic):
- Sum of zeroes = –b/a
- Product of zeroes = c/a
Example:
For x² – 5x + 6
Zeroes = 2, 3
V. Deleted Portions (CBSE 2025–2026)
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)
| Unit/Chapter | Estimated Marks | Type of Questions Typically Asked |
|---|---|---|
| Polynomials | 6–8 marks | MCQs, short answer, case-based, graph questions |
Note: This is an estimate. Actual marks may vary.
VII. Previous Year Questions (PYQs)
1 Mark
- Find degree of polynomial (CBSE 2021)
2/3 Marks
- Find zeroes of quadratic polynomial (CBSE 2020)
- Verify factor theorem (CBSE 2019)
5 Marks
- Verify relationship between zeroes and coefficients (CBSE 2018)
- Factorize using factor theorem (CBSE 2022)
VIII. Real-World Application Examples
- Used in engineering calculations
- Helps in physics equations
- Used in economics modeling
- Applied in computer graphics & animations
IX. Student Tips & Strategies for Success
Time Management
- Practice daily algebra problems (20–30 min)
- Revise formulas regularly
Exam Preparation
- Focus on concept clarity
- Practice NCERT + exemplar questions
Stress Management
- Take short breaks
- Avoid last-minute cramming
X. Career Guidance & Exploration (Class 9 Level)
Streams After Class 10:
- Science → Engineering, Medical, Research
- Commerce → Business, CA, Banking
- Arts → Law, Civil Services, Humanities
Entrance Exams (Basic Awareness):
- JEE
- NEET
- CUET
XI. Important Notes
- Always refer to NCERT textbook first
- Practice diagrams and graphs
- Revise regularly
- Focus on understanding, not memorizing
- Check updates from CBSE official website