ALGEBRAIC IDENTITIES
A Complete Reference with Proofs, Diagrams & Examples
Figure 1 — Geometric proof of (a+b)²
This document contains 12 essential algebraic identities numbered and indexed for easy reference.
Each entry includes a formal proof, a worked example, and where applicable a geometric diagram.
Prepared as a comprehensive study reference.
, Index of Identities
# Identity Name Formula
1 Square of a Sum (a + b)² = a² + 2ab + b²
2 Square of a Difference (a − b)² = a² − 2ab + b²
3 Difference of Two Squares a² − b² = (a + b)(a − b)
4 a² + b² cannot be factored over ■
Sum of Two Squares (no real factorisation)
5 Cube of a Sum (a + b)³ = a³ + 3a²b + 3ab² + b³
6 Cube of a Difference (a − b)³ = a³ − 3a²b + 3ab² − b³
7 Sum of Two Cubes a³ + b³ = (a + b)(a² − ab + b²)
8 Difference of Two Cubes a³ − b³ = (a − b)(a² + ab + b²)
9 Perfect Square Trinomial a² ± 2ab + b² = (a ± b)²
10 Product of Two Binomials (FOIL) (a + b)(c + d) = ac + ad + bc + bd
11 AM-GM Inequality (algebraic form) (a + b)² ≥ 4ab for all real a, b
12 Binomial Theorem (general) (a + b)^n = Σ C(n,k) a^(n-k) b^k, k=0..n
A Complete Reference with Proofs, Diagrams & Examples
Figure 1 — Geometric proof of (a+b)²
This document contains 12 essential algebraic identities numbered and indexed for easy reference.
Each entry includes a formal proof, a worked example, and where applicable a geometric diagram.
Prepared as a comprehensive study reference.
, Index of Identities
# Identity Name Formula
1 Square of a Sum (a + b)² = a² + 2ab + b²
2 Square of a Difference (a − b)² = a² − 2ab + b²
3 Difference of Two Squares a² − b² = (a + b)(a − b)
4 a² + b² cannot be factored over ■
Sum of Two Squares (no real factorisation)
5 Cube of a Sum (a + b)³ = a³ + 3a²b + 3ab² + b³
6 Cube of a Difference (a − b)³ = a³ − 3a²b + 3ab² − b³
7 Sum of Two Cubes a³ + b³ = (a + b)(a² − ab + b²)
8 Difference of Two Cubes a³ − b³ = (a − b)(a² + ab + b²)
9 Perfect Square Trinomial a² ± 2ab + b² = (a ± b)²
10 Product of Two Binomials (FOIL) (a + b)(c + d) = ac + ad + bc + bd
11 AM-GM Inequality (algebraic form) (a + b)² ≥ 4ab for all real a, b
12 Binomial Theorem (general) (a + b)^n = Σ C(n,k) a^(n-k) b^k, k=0..n
