Chapter 3 of 9

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Tamil Nadu (TNBSE)Class 9 Mathematics★ 10-14 marks in the TN Class 9 annual exam through MCQs, identities, theorem and equation problems

Algebra

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 3, Algebra. Polynomials, algebraic identities and factorisation, the remainder and factor theorems, GCD and LCM of polynomials, simultaneous linear equations, and linear inequations in two variables. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 24 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Apply algebraic identities to expand and factorise
2Use the remainder and factor theorems
3Find the GCD and LCM of polynomials
4Solve simultaneous linear equations
5Represent linear inequations graphically

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Why this chapter matters

Algebra is the largest and most heavily weighted chapter in TN Class 9 Maths. Identities, the remainder and factor theorems and simultaneous equations are core skills tested every year and the foundation for Class 10 algebra.

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Class 8 — Algebraic expressions

Expressions, terms and basic identities.

Algebra — Class 9 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 9 Mathematics, Chapter 3. Polynomials, identities, theorems and equations.

1. About this chapter

This chapter covers polynomials, algebraic identities, factorisation, the remainder and factor theorems, GCD/LCM of polynomials, simultaneous linear equations, and linear inequations in two variables.

2. Polynomials and identities

A polynomial in x is an expression like aₙxⁿ + … + a₁x + a₀; its degree is the highest power.
Key identities:
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

3. Remainder and factor theorems

Remainder theorem: when p(x) is divided by (x − a), the remainder is p(a).
Factor theorem: (x − a) is a factor of p(x) iff p(a) = 0.
GCD–LCM: for polynomials, GCD × LCM = product of the polynomials.

4. Equations and inequations

Simultaneous linear equations in two variables: solve by substitution, elimination, or cross-multiplication.
Linear inequations (e.g., ax + by ≤ c) are represented by a half-plane on the graph.

5. Worked examples

Example 1. Expand (2x + 3)². = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9.

Example 2. Find the remainder when p(x) = x³ − 2x + 4 is divided by (x − 1). p(1) = 1 − 2 + 4 = 3.

Example 3. Is (x − 2) a factor of p(x) = x² − 5x + 6? p(2) = 4 − 10 + 6 = 0 → yes, it is a factor.

6. Common mistakes

Mistake: Forgetting the middle term in (a + b)². Fix: (a + b)² = a² + 2ab + b².
Mistake: Using p(−a) in the remainder theorem for (x − a). Fix: For (x − a) use p(a); for (x + a) use p(−a).
Mistake: Treating an inequation like an equation when graphing. Fix: An inequation gives a region (half-plane), not just a line.

7. Practice (book-back style)

Expand (x − 4)².
Find the remainder when x³ + 3x² − 4 is divided by (x + 1).
Is (x + 3) a factor of x² + x − 6?
Factorise x² − 9.
State the factor theorem.

8. Answer key

x² − 8x + 16.
p(−1) = −1 + 3 − 4 = −2.
p(−3) = 9 − 3 − 6 = 0 → yes.
(x + 3)(x − 3).
(x − a) is a factor of p(x) if and only if p(a) = 0.

9. Quick revision

Chapter 3 · polynomials, identities, theorems, equations.
(a + b)² = a² + 2ab + b²; (a + b)(a − b) = a² − b².
Remainder theorem: remainder of p(x) ÷ (x − a) is p(a).
Factor theorem: (x − a) is a factor iff p(a) = 0; GCD × LCM = product.
Solve simultaneous equations by substitution/elimination/cross-multiplication.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Square identities

(a ± b)² = a² ± 2ab + b²

Watch the middle term.

Difference of squares

(a + b)(a − b) = a² − b²

Common factorisation.

Remainder theorem

remainder of p(x) ÷ (x − a) = p(a)

For (x + a) use p(−a).

Factor theorem

(x − a) is a factor ⟺ p(a) = 0

Tests for factors.

Polynomial GCD–LCM

GCD × LCM = product of polynomials

Like numbers.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting the middle term in (a + b)²

✓ (a + b)² = a² + 2ab + b².

WATCH OUT

✗ Using p(−a) for (x − a) in the remainder theorem

✓ For (x − a) use p(a); for (x + a) use p(−a).

WATCH OUT

✗ Graphing an inequation as a line only

✓ An inequation gives a region (half-plane).

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on identities, theorems and equations.

Questions

Identities / factorisation

Expand and factorise using identities.

Questions

Theorem / equation problems

Remainder/factor theorems and simultaneous equations.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Identity

Expand (2x + 3)².

▶ Show solution

4x² + 12x + 9.

Q2MEDIUM· Remainder

Find the remainder when p(x) = x³ − 2x + 4 is divided by (x − 1).

▶ Show solution

p(1) = 1 − 2 + 4 = 3.

Q3MEDIUM· Factor

Is (x − 2) a factor of p(x) = x² − 5x + 6?

▶ Show solution

p(2) = 4 − 10 + 6 = 0 → yes, it is a factor.

Q4EASY· Factorise

Factorise x² − 9.

▶ Show solution

(x + 3)(x − 3).

Q5MEDIUM· Remainder

Find the remainder when x³ + 3x² − 4 is divided by (x + 1).

▶ Show solution

p(−1) = −1 + 3 − 4 = −2.

Q6EASY· Concept

State the factor theorem.

▶ Show solution

(x − a) is a factor of p(x) if and only if p(a) = 0.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 3 of Samacheer Kalvi Class 9 Mathematics (largest chapter).
•(a + b)² = a² + 2ab + b²; (a + b)(a − b) = a² − b².
•Remainder theorem: remainder of p(x) ÷ (x − a) is p(a).
•Factor theorem: (x − a) is a factor iff p(a) = 0.
•GCD × LCM = product of the polynomials.
•Solve simultaneous equations by substitution/elimination/cross-multiplication.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-14 marks across MCQ, identity, theorem and equation problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-3	Identities and theorems
Short Answer	2-3	2-3	Remainder/factor and factorisation
Long Answer	5	1	Simultaneous equations / GCD-LCM

Prep strategy

Memorise the standard identities
Practise remainder and factor theorem problems
Drill simultaneous-equation methods
Learn polynomial GCD and LCM

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Problem solving

Simultaneous equations solve mixture, age and cost problems.

Engineering

Polynomials model curves and signals.

Economics

Inequations describe budget and resource constraints.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Use the correct identity for expansion
Match (x − a) with p(a) in the theorems
Show each step of equation solving
Shade the correct region for inequations

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Factorise a cubic using the factor theorem.
Solve a system of three linear equations.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN Class 9 Annual ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

By testing values of x: if p(a) = 0 then (x − a) is a factor, so you can find one factor and divide to factorise the rest of the polynomial.

When the two lines are parallel (same slope, different intercepts), the equations are inconsistent and have no common solution.

Practice more, ask doubts, find a tutor

Tamil Nadu (TNBSE) Class 9 mock tests

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