Exploring Algebraic Identities

Chapter at a glance

Identities are equations true for every value of the variables. The chapter visualises them with algebra tiles and areas, uses them to factorise and simplify, and to calculate quickly.

What this chapter covers

• 4.1 Introduction
• 4.2 Visualising Identities
• 4.4 More Identities
• 4.5 Factorisation Using Algebra Tiles
• 4.6 Factorisation Without Using Algebra Tiles
• 4.7 Finding New Identities
• 4.8 Simplifying Rational Expressions

Key points

• (x ± y)² = x² ± 2xy + y²; (x + y)(x − y) = x² − y²; (x + a)(x + b) = x² + (a + b)x + ab.
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx.
• (x ± y)³ = x³ ± 3x²y + 3xy² ± y³; x³ ± y³ = (x ± y)(x² ∓ xy + y²).
• x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx).
• Rational expressions are simplified by factorising and cancelling common non-zero factors.

Read the NCERT chapter

Read this chapter in the official NCERT textbook Ganita Manjari (2026-27 edition) for free on ncert.nic.in.

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