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I. Chapter Summary
This chapter builds on your understanding of algebraic expressions and focuses on polynomials, particularly quadratic polynomials. It introduces the relationship between zeroes and coefficients, the geometrical meaning of zeroes, and the division algorithm for polynomials. Understanding these concepts is essential for progressing in algebra and calculus.
II. Key Concepts Covered
| Concept | Description |
|---|---|
| Polynomial | An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. |
| Types of Polynomials | Linear: Degree 1 (e.g., $3x + 1$) Quadratic: Degree 2 (e.g., $x^2 + 3x + 2$) Cubic: Degree 3 (e.g., $x^3 – 2x + 1$) |
| Zeroes of a Polynomial | The values of the variable for which the polynomial becomes zero. |
| Geometrical Meaning of Zeroes | The x-intercepts of the graph of the polynomial. |
| Relationship between Zeroes and Coefficients | For a quadratic polynomial $ax^2 + bx + c$: Sum of zeroes = $-\frac{b}{a}$ Product of zeroes = $\frac{c}{a}$ |
| Division Algorithm for Polynomials | For polynomials $p(x)$ and $g(x)$, the division algorithm is given by: $p(x) = g(x) \cdot q(x) + r(x)$, where $r(x) = 0$ or $\deg(r(x)) < \deg(g(x))$. |
III. Important Questions
(A). Multiple Choice Questions (1 Mark)
Q1. What is the degree of the polynomial $3×4−2×2+x−73x^4 – 2x^2 + x – 73×4−2×2+x−7?$
a) 2
b) 3
c) 4 ✅
d) 1
Q2. If α and β are the zeroes of the polynomial $x2−5x+6x^2 – 5x + 6×2−5x+6, then $α + β = ?$
a) 5 ✅ (PYQ 2020)
b) 6
c) −5
d) −6
Q3. The number of zeroes a quadratic polynomial can have is:
a) 0
b) 1
c) 2 ✅ (PYQ 2021)
d) 3
Q4. What is the zero of the polynomial $x−4x – 4x−4?$
a) 0
b) 4 ✅
c) −4
d) 1
(B). Short Answer Questions (2/3 Marks)
1. Find the zeroes of the polynomial $x2+5x+6x^2 + 5x + 6×2+5x+6$ and verify the relationship between the zeroes and coefficients.
2. Given that 2 is a zero of the polynomial $f(x)=x2−3x+kf(x) = x^2 – 3x + kf(x)=x2−3x+k$, find the value of k.
3. Use the division algorithm to divide $p(x)=x3−3×2+5x−3p(x) = x^3 – 3x^2 + 5x – 3p(x)=x3−3×2+5x−3 by g(x)=x−1g(x) = x – 1g(x)=x−1$.
4. If the zeroes of the quadratic polynomial $f(x)=x2+px+qf(x) = x^2 + px + qf(x)=x2+px+q$ are equal, show that $p2=4qp^2 = 4qp2=4q$.
(C). Long Answer Questions (5 Marks)
1. Find a quadratic polynomial whose zeroes are 3 and −4. Also verify the relation between the zeroes and the coefficients.Divide
2. $p(x)=x3−6×2+11x−6p(x) = x^3 – 6x^2 + 11x – 6p(x)=x3−6×2+11x−6 by x−1x – 1x−1$ and verify division algorithm.
3. If α and β are the zeroes of the polynomial $2×2+3x+72x^2 + 3x + 72×2+3x+7$, find a quadratic polynomial whose zeroes are $(1/α)$ and $(1/β)$.
4. Find the zeroes of the polynomial $f(x)=x2−4x−5f(x) = x^2 – 4x – 5f(x)=x2−4x−5$ and verify the relationship between zeroes and coefficients. (PYQ 2019)
(D). HOTS (Higher Order Thinking Skills)
1. Construct a polynomial whose zeroes are the squares of the zeroes of the polynomial $x2−3x+2x^2 – 3x + 2×2−3x+2$.
2. If one zero of the polynomial $ax2+bx+cax^2 + bx + cax2+bx+c$ is double the other, find the relationship between a, b, and c.
IV. Key Formulas/Concepts
| Formula/Concept | Description |
| Zero of Polynomial | If $f(a)=0f(a) = 0f(a)=0$, then a is a zero of the polynomial. |
| Sum and Product of Zeroes (Quadratic) | Sum = $−ba-\frac{b}{a}−ab$, |
| Product = $ca\frac{c}{a}ac$ | |
| General Quadratic Form | $ax^2 + bx + c$ where $a \ne 0$ |
| Division Algorithm | $p(x) = g(x) \cdot q(x) + r(x)$ |
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 | Types of Questions Typically Asked |
| Chapter 2: Polynomials | 6–8 Marks | 1 Long Answer, 2 Short Answers, 1–2 MCQs |
VII. Previous Year Questions (PYQs)
| Year | Marks | Question |
| 2021 | 1 | What is the maximum number of zeroes a quadratic polynomial can have? (2) |
| 2020 | 1 | Find the sum of zeroes of $x^2 – 5x + 6$ |
| 2019 | 5 | Find zeroes of $x2−4x−5x^2 – 4x – 5×2−4x−5$ and verify the relationship. |
VIII. Real-World Application Examples to Connect with Topics
| Concept | Application |
| Zeroes of Polynomial | Projectile motion in physics, revenue and cost analysis in economics |
| Graphical Meaning of Zeroes | Used in data visualization, machine learning, and engineering curves |
| Polynomial Division | Algorithmic logic in computer programs and coding |
| Quadratic Modelling | Height-time relationships, business profit modeling |
IX. Student Tips & Strategies for Success
Time Management
- Daily practice of 2–3 polynomial problems.
- Focus on pattern-based problems: factorization, root verification, polynomial construction.
Exam Preparation
- Practice PYQs and NCERT exemplar problems.
- Ensure understanding of relation between zeroes and coefficients.
Stress Management
- Use visual aids like graphs for understanding polynomial behavior.
- Create flashcards of key formulas for daily revision.
X. Career Guidance & Exploration (Class-Specific)
For Classes 9–10
Streams
- Science: Physics, Computer Science, Engineering
- Commerce: Economics, Market Analytics
- Arts: Architecture, Game Design
Career Options
- Data Scientist, Algorithm Designer, Economist, Business Analyst
Exams
- NTSE, IMO, RMO, KVPY, CUET Aptitude Sections
XI. Important Notes
- Always refer to https://cbseacademic.nic.in and https://ncert.nic.in for official updates.
- Mastering this chapter enhances your foundation for Algebra, Coordinate Geometry, and future calculus.
- Focus on conceptual clarity over memorization, especially in constructing polynomials and verifying results.