Algebraic Identities
An algebraic identity is an equation that holds true for all values of the variables involved. Unlike an algebraic equation, which is true only for specific values of the unknown, an identity is universally valid — it is essentially a statement about the structure of algebraic expressions rather than a puzzle to solve. For example, (a + b)² = a² + 2ab + b² is true whether a = 1 and b = 2, or a = −5 and b = 100, or any other combination of real numbers. Algebraic identities are among the most powerful tools in a mathematician's toolkit. They allow us to expand products of polynomials quickly, factorise complicated expressions into simpler factors, and simplify calculations that would otherwise be very tedious. In competitive mathematics and everyday algebra alike, recognising the 'shape' of an expression and matching it to a known identity is a skill that dramatically speeds up problem-solving. The standard identities studied at the high-school level fall into a small set of memorable patterns: the square of a sum, the square of a difference, the difference of two squares, the cube of a sum, the cube of a difference, and the sum or difference of two cubes. Mastering these six groups provides a solid foundation for factorisation, simplification of rational expressions, and later work in calculus and number theory.
- Square of a Sum
- Square of a Difference
- Difference of Two Squares
- Cube of a Sum
- Cube of a Difference
- Sum and Difference of Two Cubes
Square of a Sum
Identity: (a + b)² = a² + 2ab + b². Proof by expansion: (a + b)(a + b) = a·a + a·b + b·a + b·b = a² + 2ab + b². Example use: expand (3x + 5)² = 9x² + 30x + 25.
Square of a Difference
Identity: (a − b)² = a² − 2ab + b². Proof: (a − b)(a − b) = a² − ab − ba + b² = a² − 2ab + b². Example: (4x − 7)² = 16x² − 56x + 49. Note that (a − b)² ≥ 0 for all real a, b, since it is a perfect square.
Difference of Two Squares
Identity: (a + b)(a − b) = a² − b². This is perhaps the most-used identity in factorisation. Example: factor 25x² − 16 = (5x)² − 4² = (5x + 4)(5x − 4). It also gives a neat mental-arithmetic trick: 63 × 57 = (60 + 3)(60 − 3) = 3600 − 9 = 3591.
Cube of a Sum
Identity: (a + b)³ = a³ + 3a²b + 3ab² + b³. This follows from multiplying (a + b)² by (a + b): (a² + 2ab + b²)(a + b) = a³ + a²b + 2a²b + 2ab² + ab² + b³ = a³ + 3a²b + 3ab² + b³. The coefficients 1, 3, 3, 1 appear in row 3 of Pascal's triangle.
Cube of a Difference
Identity: (a − b)³ = a³ − 3a²b + 3ab² − b³. Replace b with −b in the cube-of-sum identity to obtain this. Example: (2x − 1)³ = 8x³ − 12x² + 6x − 1.
Sum and Difference of Two Cubes
Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²). Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²). Verify sum of cubes by expanding: (a + b)(a² − ab + b²) = a³ − a²b + ab² + a²b − ab² + b³ = a³ + b³. ✓ Example: factor 8x³ − 27 = (2x)³ − 3³ = (2x − 3)(4x² + 6x + 9).
The Key Equation
- a — first term (dimensionless): Any real number or algebraic expression representing the first quantity.
- b — second term (dimensionless): Any real number or algebraic expression representing the second quantity.
- 2ab — cross-product term (dimensionless): Twice the product of a and b; this term is what distinguishes (a+b)² from a²+b² and is the most commonly forgotten part.
Worked Example
Problem. Factorise completely: 9x² − 30x + 25 − 4y²
Answer. (3x + 2y − 5)(3x − 2y − 5)
Origins · 1 minBrahmagupta and how this was discovered
Brahmagupta was a 7th-century Indian mathematician and astronomer who made foundational contributions to algebra. In his landmark work Brāhmasphuṭasiddhānta (628 CE), he systematically used algebraic identities and was among the first to give rules for operating with zero and negative numbers. His work on quadratic equations and identities such as the Brahmagupta–Fibonacci identity (expressing a product of sums of two squares as itself a sum of two squares) directly influenced later algebraic development in the Islamic world and Europe. Brahmagupta ran a school at Bhillamala (modern Bhinmal, Rajasthan) and wrote his major mathematical treatise at the age of just 30. He is also credited with one of the earliest explicit treatments of zero as a number in its own right.
Related Concepts
- Factorisation of Polynomials — Algebraic identities are the primary tool for factorising polynomials; recognising the pattern of an identity reveals the factor pairs directly.
- Expansion of Polynomials — Identities are used in reverse during expansion, turning compact bracket forms into fully expanded polynomial expressions.
- Pascal's Triangle — The coefficients in the cube-of-sum identity (1, 3, 3, 1) come directly from Pascal's triangle, which generalises to the binomial theorem for any power.
- Completing the Square — The square-of-a-sum and square-of-a-difference identities underlie the technique of completing the square, used to solve quadratic equations and find vertex form of parabolas.