Chapter 2 · Class X Mathematics · NCERT Exercises

Polynomials — Exercises

Zeroes, Graphs & Division — All Polynomial Exercises Solved

📂 4 Exercises 📝 13 Questions 🎓 Moderate

▶ Start Exercises 📖 Revise Notes

Exercise Index

Geometrical meaning of zeroes; graphs of polynomials

Relationship between zeroes and coefficients; form poly from zeroes

Division algorithm for polynomials; factorisation

5 Q Moderate-Hard

Mixed — verify zeroes, division, construct polynomials

4 exercise files · 13 total questions

Chapter at a Glance

CBSE BoardsNTSEJEE Foundation

8 Concepts

6 Formulas

Moderate Difficulty

5–7% Weightage

Before You Begin

Prerequisites

✓Algebraic identities (Class IX)
✓Factorisation of quadratics
✓Graph plotting basics

Have Ready

🔧Graph paper for parabola sketches
🔧Long division layout template

Exercise Topic Map

Exercise 2.1 Count x-intercepts of graph = number of real zeroes; parabola orientation from leading coefficient

Exercise 2.2 α+β = −b/a; αβ = c/a; αβγ = −d/a for cubics; build p(x) = x²−(α+β)x+αβ

Exercise 2.3 Divide p(x) by g(x): p(x) = g(x)·q(x)+r(x); verify using remainder

Miscellaneous Combine all skills: find zeroes, form polynomial, verify division algorithm

Key Formulae — Recall Before Solving

\(\alpha + \beta = -\dfrac{b}{a},\quad \alpha\beta = \dfrac{c}{a} \quad \text{(quadratic)}\)

\(\alpha+\beta+\gamma = -\dfrac{b}{a},\quad \alpha\beta\gamma = -\dfrac{d}{a} \quad \text{(cubic)}\)

\(p(x) = g(x) \cdot q(x) + r(x),\quad \deg r < \deg g\)

\(p(\alpha)=0 \iff \alpha \text{ is a zero of } p(x)\)

NCERT Solving Method

Step 1 — Zeroes from graph: x-coordinates where curve cuts x-axis. Step 2 — Coefficient relations: memorise α+β = −b/a and αβ = c/a; never mix up signs. Step 3 — Polynomial from zeroes: p(x) = k[x² − (sum)x + (product)]; k = 1 unless stated. Step 4 — Division algorithm: do polynomial long division carefully, subtract at each step; check p(x) = g(x)q(x)+r(x) at the end.

Continue Your Preparation

📖NotesTheory & concepts 🧠MCQsTopic-wise practice ⚖️True–FalseConcept check ➡️Next ChapterPair of Linear Equations

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