Algebraic Identity

Algebraic Identity

In algebra, an equality that is true regardless of the numerical values assigned to its variables.

Symbol

The symbol “≡” is sometimes used for trigonometric identities.

Properties

These are special algebraic identities :

  • \({(a+b)^2}\) = \({a^2}\)+ 2ab + \({b^2}\)
  • \({(a–b)^2}\) = \({a^2}\)– 2ab + \({b^2}\)
  • \({(a–b)^3}\) = \({a^3}\)– 3a\({b^2}\)+ 3\({a^2}\)b – \({b^3}\)
  • \({(a+b)^3}\) = \({a^3}\)+ 3a\({b^2}\)+ 3\({a^2}\)b + \({b^3}\)
  • \({a^2}\) –  \({b^2}\) = (a + b) (ab)
  • \({a^3}\) –  \({b^3}\) = (ab)(\({a^2}\) + ab + \({b^2}\))
  • \({a^3}\) +  \({b^3}\) = (a + b)(\({a^2}\) – ab + \({b^2}\))

Examples

  • The equality (a + b)² = a² + 2ab + b² is an algebraic identity.
  • The relation sin (A + B) ≡ sin A cos B + sin B cos A is a trigonometric identity.

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