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Rajasthan (RBSE)Class 9 Mathematics★ 6–8 marks · RBSE Class 9 Mathematics (new NCF — Ganita Prakash 9)

Exploring Algebraic Identities

The new NCF Class 9 Maths chapter on algebraic identities. Visualising and proving (a+b)², (a−b)², a²−b², (x+a)(x+b), (a+b+c)² and the cubic identities; then using identities to expand quickly and to factorise — with solved board-pattern problems.

⏱ 28 min readMedium difficulty✓ Free · NCERT-aligned

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Key points, glossary and the questions most likely to show up.

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

1Distinguish an identity (true for all values) from an equation
2State and visualise the square identities geometrically
3Use identities to expand expressions quickly
4Use identities to factorise (difference of squares, sum/difference of cubes)
5Apply identities to evaluate clever numerical products and rearrangements

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Why this chapter matters

Algebraic identities are the most reused tools in all of maths — expanding, factorising, simplifying and evaluating clever products all rely on them. They are guaranteed marks in the RBSE paper and the foundation for Class 10 Polynomials and Quadratic Equations.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Introduction to Linear Polynomials (Class 9)

Terms, coefficients and basic factorisation.

Exploring Algebraic Identities (RBSE Class 9 · Mathematics)

An identity is an equation that is true for every value of the variables — not just a lucky few. Once you know a handful of them, you can expand and factorise huge expressions in a single line. The new book even shows you how to see them as areas of squares and rectangles.

RBSE note (2026-27). Class 9 uses the new NCF (Ganita Prakash 9) Mathematics textbook. Exploring Algebraic Identities follows the polynomials chapter. BSER (Ajmer) sets the exam.

1. Identity vs equation

An equation (e.g. 2x + 1 = 5) is true only for particular values of x.
An identity (e.g. (a + b)² = a² + 2ab + b²) is true for all values of the variables.

You can visualise the square identities as areas: a square of side (a + b) splits into an a×a square, a b×b square and two a×b rectangles — area a² + 2ab + b².

2. The standard identities

Square identities $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ $a^{2} - b^{2} = (a + b) (a - b)$

Product and three-term $(x + a) (x + b) = x^{2} + (a + b) x + ab$ $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 ab + 2 b c + 2 c a$

Cubic identities $(a + b)^{3} = a^{3} + b^{3} + 3 ab (a + b)$ $(a - b)^{3} = a^{3} - b^{3} - 3 ab (a - b)$ $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$ $a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)$

3. Using identities to expand

Identities turn slow multiplication into a quick substitution.

Expand (2x + 3y)².

Use (a + b)² with a = 2x, b = 3y: $(2 x)^{2} + 2 (2 x) (3 y) + (3 y)^{2} = 4 x^{2} + 12 x y + 9 y^{2} .$

Evaluate 103 × 97 without multiplying directly.

103 × 97 = (100 + 3)(100 − 3) = 100² − 3² = 10000 − 9 = 9991 (using a² − b²).

4. Using identities to factorise

Read the identities right to left to factorise.

Factorise 9x² − 16y².

This is a² − b² with a = 3x, b = 4y: $9 x^{2} - 16 y^{2} = (3 x + 4 y) (3 x - 4 y) .$

Factorise x³ + 8.

This is a³ + b³ with a = x, b = 2: $x^{3} + 8 = (x + 2) (x^{2} - 2 x + 4) .$

5. Worked example

If a + b = 7 and ab = 12, find a² + b².

Use (a + b)² = a² + 2ab + b², so a² + b² = (a + b)² − 2ab: $a^{2} + b^{2} = 7^{2} - 2 (12) = 49 - 24 = 25.$

6. Quick recap

An identity holds for all values; an equation only for some.
Master the squares: $(a \pm b)^{2}$ , $a^{2} - b^{2} = (a + b) (a - b)$ , $(x + a) (x + b)$ , $(a + b + c)^{2}$ .
Master the cubics: $(a \pm b)^{3}$ , $a^{3} \pm b^{3}$ , and $a^{3} + b^{3} + c^{3} - 3 ab c$ .
Read left→right to expand, right→left to factorise.
Tricks: clever pairing (e.g. 103×97 = 100²−3²) and the rearrangement a²+b² = (a+b)² − 2ab.

Key formulas & results

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

Square of a sum

(a+b)² = a² + 2ab + b²

Visualise as a square of side (a+b).

Square of a difference

(a−b)² = a² − 2ab + b²

Difference of squares

a² − b² = (a+b)(a−b)

The most-used factorisation.

Product (x+a)(x+b)

x² + (a+b)x + ab

Basis of splitting the middle term.

Square of three terms

(a+b+c)² = a²+b²+c²+2ab+2bc+2ca

Sum/difference of cubes

a³±b³ = (a±b)(a²∓ab+b²)

Three-cube identity

a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)

Zero when a+b+c = 0.

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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

✗ Writing (a+b)² = a² + b²

✓ You must include the middle term: (a+b)² = a² + 2ab + b². Forgetting 2ab is the single most common error.

WATCH OUT

✗ Mixing the signs in a³ − b³

✓ a³ − b³ = (a − b)(a² + ab + b²): the quadratic factor has a PLUS ab. (For a³ + b³ it is − ab.)

WATCH OUT

✗ Factorising a² + b² as (a+b)(a−b)

✓ Only the DIFFERENCE of squares factorises over the reals: a² − b² = (a+b)(a−b). a² + b² does not factorise (over reals).

WATCH OUT

✗ Dropping cross-terms in (a+b+c)²

✓ There are three cross terms: +2ab +2bc +2ca. Don't forget the 2ca.

WATCH OUT

✗ Using an identity with mismatched substitution

✓ Identify a and b clearly (e.g. for 9x², a = 3x) and substitute consistently everywhere.

NCERT exercises (with solutions)

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

In-text (visualising identities)

Area models of (a+b)², (a−b)²

Questions

In-text (expand)

Apply identities to expand

Questions

In-text (factorise)

Difference of squares, cubes

Questions

End-of-chapter Ex

End-of-chapter exercise

Expand, factorise, evaluate clever products

Questions

Practice problems

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

Q1EASY· Expand

Expand (x + 5)².

▶ Show solution

Step 1 — (a+b)² with a = x, b = 5.

Step 2 — x² + 2(x)(5) + 25 = x² + 10x + 25.

✦ Answer: x² + 10x + 25.

Q2EASY· Difference of squares

Factorise x² − 49.

▶ Show solution

Step 1 — a² − b² with a = x, b = 7.

Step 2 — (x + 7)(x − 7).

✦ Answer: (x + 7)(x − 7).

Q3EASY· Clever product

Use an identity to find 102 × 98.

▶ Show solution

Step 1 — (100 + 2)(100 − 2) = 100² − 2².

Step 2 — 10000 − 4 = 9996.

✦ Answer: 9996.

Q4MEDIUM· Expand

Expand (3a − 2b)².

▶ Show solution

Step 1 — (a−b)² with a = 3a, b = 2b.

Step 2 — (3a)² − 2(3a)(2b) + (2b)² = 9a² − 12ab + 4b².

✦ Answer: 9a² − 12ab + 4b².

Q5MEDIUM· Rearrange

If a + b = 10 and ab = 21, find a² + b².

▶ Show solution

Step 1 — a² + b² = (a+b)² − 2ab.

Step 2 — = 10² − 2(21) = 100 − 42 = 58.

✦ Answer: 58.

Q6MEDIUM· Cubes

Factorise 27x³ + 1.

▶ Show solution

Step 1 — a³ + b³ with a = 3x, b = 1.

Step 2 — (3x + 1)((3x)² − (3x)(1) + 1²) = (3x + 1)(9x² − 3x + 1).

✦ Answer: (3x + 1)(9x² − 3x + 1).

Q7HARD· Three-term

Expand (x + 2y + 3z)².

▶ Show solution

Step 1 — (a+b+c)² = a²+b²+c²+2ab+2bc+2ca with a=x, b=2y, c=3z.

Step 2 — x² + 4y² + 9z² + 2(x)(2y) + 2(2y)(3z) + 2(3z)(x).

Step 3 — x² + 4y² + 9z² + 4xy + 12yz + 6zx.

✦ Answer: x² + 4y² + 9z² + 4xy + 12yz + 6zx.

Q8HARD· Three-cube

Evaluate 8³ + (−5)³ + (−3)³ using an identity.

▶ Show solution

Step 1 — here a + b + c = 8 + (−5) + (−3) = 0.

Step 2 — when a + b + c = 0, a³ + b³ + c³ = 3abc.

Step 3 — 3abc = 3 × 8 × (−5) × (−3) = 360.

✦ Answer: 360.

5-minute revision

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

•Identity = true for ALL values; equation = true for some.
•(a±b)² = a² ± 2ab + b² (never forget 2ab); a² − b² = (a+b)(a−b).
•(x+a)(x+b) = x² + (a+b)x + ab; (a+b+c)² = a²+b²+c²+2ab+2bc+2ca.
•a³ ± b³ = (a ± b)(a² ∓ ab + b²); mind the middle sign.
•a³+b³+c³−3abc = (a+b+c)(…); equals 3abc when a+b+c = 0.
•Expand left→right; factorise right→left.
•Rearrangement: a²+b² = (a+b)² − 2ab; clever products via a²−b².

Rajasthan (RBSE) marks blueprint

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

Typical chapter weightage: 6–8 marks

Question type	Marks each	Typical count	What it tests
MCQ / fill in the blank	1	1–2	Expand a square, difference of squares, clever product
Short answer	2	2	Expand binomials, rearrangement (a²+b²), cube factorisation
Short/Long answer	3	1	(a+b+c)² expansion; three-cube identity

Prep strategy

Memorise all identities and practise reading them BOTH ways (expand/factorise)
Drill the 'evaluate clever product' trick using a²−b²
Lock the rearrangement a²+b² = (a+b)² − 2ab; it appears every year
Watch cube-identity signs (+ab vs −ab) and the three cross-terms in (a+b+c)²

Where this shows up in the real world

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

Fast mental maths

Tricks like 103×97 = 100²−3² come straight from a²−b².

Engineering algebra

Expanding and factorising with identities simplifies formulas in physics and engineering.

Cryptography

Difference-of-squares factorisation is a stepping stone to integer factorisation methods.

Area and design

The geometric (area) view of (a+b)² is used in tiling and packing problems.

Exam strategy

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

Spot which identity fits BEFORE multiplying out — it saves time and errors.
For clever numericals, rewrite the numbers as (m ± n) to use a²−b² or (a±b)².
Read identities right-to-left to factorise; match a and b carefully.
Mind cube-identity signs and include all three cross-terms in (a+b+c)².
Show the identity used as your first line — it earns method marks.

Going beyond the textbook

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

Sophie Germain identity: a⁴ + 4b⁴ = (a²+2b²+2ab)(a²+2b²−2ab).
Symmetric functions and Newton's identities.
Using a+b+c = 0 ⇒ a³+b³+c³ = 3abc in competition problems.

Where else this chapter is tested

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

RBSE Class 9 Annual (BSER Ajmer)Very high — expand/factorise guaranteed

NTSE / NMMSMedium — identity-based MCQs

JEE FoundationVery high — base for Class 10 algebra

Maths Olympiad (IMO)Medium — algebraic manipulation

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Questions students ask

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

Yes. Class 9 (2026-27) uses the new NCF NCERT 'Ganita Prakash 9' book; 'Exploring Algebraic Identities' covers the square, product and cubic identities and their use in expanding and factorising. BSER Ajmer sets the RBSE paper.

Squaring a sum produces the cross term: (a+b)(a+b) = a² + ab + ba + b² = a² + 2ab + b². The 2ab comes from the two middle products and cannot be dropped.

Exactly when a + b + c = 0. Then the factor (a+b+c) is zero, so a³+b³+c³ − 3abc = 0, giving a³+b³+c³ = 3abc.

Not over the real numbers. Only the DIFFERENCE of squares factorises: a² − b² = (a+b)(a−b).

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