Algebraic identities are standard mathematical formulas that hold true for all variable values. They are commonly used to simplify, expand, or factor algebraic expressions without direct computation and are essential in solving equations and proving other algebraic results.
\[ (a \pm b)^2 = a^2 \pm 2ab + b^2 \]
\[ (a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3 \]
\[ (a \pm b)^4 = a^4 \pm 4a^3b + 6a^2b^2 \pm 4ab^3 + b^4 \]
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \]
\[ (a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc \]
\[ (a - b - c)^2 = a^2 + b^2 + c^2 - 2ab - 2ac + 2bc \]
\[ (a + b + c)^3 = a^3 + b^3 + c^3 + 6abc + 3(a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2) \]
\[ \left(a_1 + a_2 + \dots + a_n\right)^2 = \sum_{i=1}^n a_i^2 + 2 \sum_{i < j} a_ia_j \]
\[ a^2 - b^2 = (a - b)(a + b) \]
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
\[ a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 \]
\[ a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \]
\[ a^5 + b^5 = (a + b)(a^4 - a^3b + a^2b^2 - ab^3 + b^4) \]
\[ a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) \]