# Algebraic Identities

An **identity** is an equality that holds true regardless of the values chosen for its variables. They are used in simplifying or rearranging algebra expressions. By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time.

( x + y )^{2} |
= | x^{2} + 2 x y + y^{2} |

( x - y )^{2} |
= | x^{2} - 2 x y + y^{2} |

( x + y )^{3} |
= | x^{3 }+ 3 x^{2} y + 3 x y^{2} + y^{3} |

( x - y )^{3} |
= | x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3} |

( x + y )^{4} |
= | x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} |

( x - y )^{4} |
= | x^{4} - 4 x^{3} y + 6 x^{2} y^{2} - 4 x y^{3} + y^{4} |

x^{2} - y^{2} |
= | ( x + y ) ( x - y ) |

x^{3 }- y^{3} |
= | ( x - y ) ( x^{2} + x y + y^{2 }) |

x^{3} + y^{3} |
= | ( x + y ) ( x^{2} - x y + y^{2 }) |

x^{4} - y^{4} |
= | ( x^{2} - y^{2 }) ( x^{2} + y^{2} ) |