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Schooling · Class 9 · Mathematics · Exploring Algebraic Identities · Chapter on a page

CBSE Class 9 · Mathematics · NCERT Ganita Manjari

Exploring Algebraic Identities: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Exploring Algebraic Identities: Key ideas, Formulas and facts, Watch out forKey ideasAn algebraic identity is true for…The square identities for two terms…The difference of squares identity…The cube identities areThe sum and difference of cubesIdentities can be verified by…Factorisation is the reverse of…Formulas and facts(a + b)² = a² + 2ab + b² — square of…(a - b)² = a² - 2ab + b² — square of…a² - b² = (a + b)(a - b)…(a + b)³ = a³ + 3a²b + 3ab² + b³…(a - b)³ = a³ - 3a²b + 3ab² - b³…a³ + b³ = (a + b)(a² - ab + b²)…a³ - b³ = (a - b)(a² + ab + b²)…Watch out forForgetting the middle term in (a +…Confusing signs in (a - b)³…Mixing up sum of cubes and…Applying (a + b)² to (a + b)³ by…Forgetting to square the…Exploring AlgebraicIdentities
The mind map as a list
  • Key ideas
    • An algebraic identity is true for…
    • The square identities for two terms…
    • The difference of squares identity…
    • The cube identities are
    • The sum and difference of cubes
    • Identities can be verified by…
    • Factorisation is the reverse of…
  • Formulas and facts
    • (a + b)² = a² + 2ab + b² — square of…
    • (a - b)² = a² - 2ab + b² — square of…
    • a² - b² = (a + b)(a - b)…
    • (a + b)³ = a³ + 3a²b + 3ab² + b³…
    • (a - b)³ = a³ - 3a²b + 3ab² - b³…
    • a³ + b³ = (a + b)(a² - ab + b²)…
    • a³ - b³ = (a - b)(a² + ab + b²)…
  • Watch out for
    • Forgetting the middle term in (a +…
    • Confusing signs in (a - b)³…
    • Mixing up sum of cubes and…
    • Applying (a + b)² to (a + b)³ by…
    • Forgetting to square the…

What this chapter is about

Algebraic identities are equations that hold true for all values of the variables involved. Unlike ordinary equations that are true only for specific values, an identity like (a + b)² = a² + 2ab + b² works no matter what numbers you substitute for a and b. This chapter builds on the basic identities you learned in earlier classes and extends them to more powerful forms.

Key ideas

  • An algebraic identity is true for every value of its variables, while an equation is true only for certain values.
  • The square identities for two terms are: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b².
  • The difference of squares identity states: a² - b² = (a + b)(a - b).
  • The cube identities are: (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a - b)³ = a³ - 3a²b + 3ab² - b³.
  • The sum and difference of cubes: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²).
  • Identities can be verified by expanding both sides and showing they are equal, or by substituting numerical values.
  • Factorisation is the reverse of expansion: you start with an expanded form and write it as a product using identities.

Formulas and facts to remember

  • (a + b)² = a² + 2ab + b² — square of a sum equals sum of squares plus twice the product.
  • (a - b)² = a² - 2ab + b² — square of a difference equals sum of squares minus twice the product.
  • a² - b² = (a + b)(a - b) — difference of two squares factors into sum times difference.
  • (a + b)³ = a³ + 3a²b + 3ab² + b³ — cube of a sum expands to four terms.
  • (a - b)³ = a³ - 3a²b + 3ab² - b³ — cube of a difference, note the alternating signs.
  • a³ + b³ = (a + b)(a² - ab + b²) — sum of cubes factors with a minus in the second bracket.
  • a³ - b³ = (a - b)(a² + ab + b²) — difference of cubes factors with a plus in the second bracket.
  • (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca — square of a trinomial.

Common mistakes

  • Forgetting the middle term in (a + b)², writing it as a² + b² instead of a² + 2ab + b² → always include 2ab.
  • Confusing signs in (a - b)³, especially the sign of the last term → write out each term carefully and track the minus.
  • Mixing up sum of cubes and difference of cubes factorisations → remember: a³ + b³ has (a + b) with a minus inside the trinomial; a³ - b³ has (a - b) with a plus inside.
  • Applying (a + b)² to (a + b)³ by mistake → cube identities have four terms, not three.
  • Forgetting to square the coefficients when expanding, writing (2x)² as 2x² instead of 4x² → square both the number and the variable part.

Quick revision

  • (a + b)² and (a - b)² always have three terms; the middle term is 2ab.
  • a² - b² splits into two brackets: one with a plus, one with a minus.
  • Cube identities have four terms; sum/difference of cubes factor into a binomial times a trinomial.
  • To use identities for arithmetic, express numbers as convenient sums or differences near round figures.
  • Factorisation is expansion done backwards—spot the pattern first.