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.
In Class 9, you explore identities involving squares, cubes, and products of binomials. These identities help you expand expressions quickly, factorise polynomials, and simplify calculations. Many problems in algebra, coordinate geometry, and later in quadratic equations depend on recognising and applying these identities.
After studying this chapter, you should be able to expand expressions using standard identities, factorise polynomials by identifying which identity applies, and use identities to perform mental arithmetic with large numbers.
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.
Worked examples
Example 1: Expand (3x + 4y)²
Using the identity (a + b)² = a² + 2ab + b², let a = 3x and b = 4y.
(3x + 4y)² = (3x)² + 2 × (3x) × (4y) + (4y)²
= 9x² + 24xy + 16y²
Example 2: Factorise 27m³ - 8
Notice that 27m³ = (3m)³ and 8 = 2³. This is a difference of cubes.
Using a³ - b³ = (a - b)(a² + ab + b²), let a = 3m and b = 2.
27m³ - 8 = (3m - 2)((3m)² + (3m)(2) + 2²)
= (3m - 2)(9m² + 6m + 4)
Example 3: Find the value of 103² using an identity
Write 103 as 100 + 3. Use (a + b)² where a = 100 and b = 3.
103² = (100 + 3)²
= 100² + 2 × 100 × 3 + 3²
= 10000 + 600 + 9
= 10609
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.