Chapter 2 · Class IX Mathematics · NCERT Exercises

Polynomials — Exercises

Zeroes, Remainder Theorem & Algebraic Identities — All 4 Exercises Solved

📂 4 Exercises 📝 17 Questions 🎓 Moderate

▶ Start Exercises 📖 Revise Notes

Exercise Index

Polynomials in one variable; degree; coefficients; types: monomial, binomial, trinomial

Zeroes of a polynomial; value of polynomial at a point; verify zeroes

4 Q Easy-Moderate

Remainder Theorem; Factor Theorem; factorisation using factor theorem

Algebraic identities; expansion using identities; factorisation; (a+b+c)², (a±b)³, a³±b³

5 Q Moderate-Hard

4 exercise files · 17 total questions

Chapter at a Glance

CBSE BoardsNTSEJEE Foundation

10 Concepts

8 Formulas

Moderate Difficulty

8–10% Weightage

Before You Begin

Prerequisites

✓Algebraic expressions (Class VIII)
✓Factorisation basics
✓Substitution in expressions

Have Ready

🔧Identity expansion chart
🔧Rough paper for long division of polynomials

Exercise Topic Map

Exercise 2.1 Identify degree (highest power); check if expression is polynomial (no negative/fractional exponents on variable)

Exercise 2.2 Substitute the value into p(x); if p(a) = 0, then a is a zero; find zeroes by inspection or factoring

Exercise 2.3 Remainder when p(x) ÷ (x−a) = p(a); Factor Theorem: (x−a) is factor ↔ p(a)=0

Exercise 2.4 Memorise all 9 identities; for complex factorisations: try grouping or substitution first

Key Formulae — Recall Before Solving

\(p(a) = 0 \iff (x-a) \text{ is a factor of } p(x) \quad \text{(Factor Theorem)}\)

\((a+b)^2 = a^2+2ab+b^2;\quad (a-b)^2 = a^2-2ab+b^2;\quad a^2-b^2 = (a+b)(a-b)\)

\((a+b)^3 = a^3+3a^2b+3ab^2+b^3;\quad a^3+b^3 = (a+b)(a^2-ab+b^2)\)

\((a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca\)

NCERT Solving Method

Step 1 — Zeroes: substitute the given value; simplify step-by-step; confirm result is 0. Step 2 — Remainder Theorem: just evaluate p(a); no need to actually divide. Step 3 — Factor Theorem: test p(1), p(−1), p(2), p(−2) etc. to find a zero; use it as a factor; do polynomial long division for remaining factor. Step 4 — Identity-based factorisation: recognise the structure; match with the standard identity; factorise accordingly. Step 5 — (a+b+c)² expansion: expand carefully; remember 2ab, 2bc, 2ca cross terms.

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📖NotesTheory & concepts 🧠MCQsTopic-wise practice ⚖️True–FalseConcept check ➡️Next ChapterCoordinate Geometry

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