Algebraic identities

An equality property between two algebraic expressions is called an algebraic identity.

Introduction

In algebra, two expressions in algebraic form are equal. The mathematical relationship between them is called an algebraic identity. There are some useful algebraic identities and they are used as formulas in mathematics. The following is the list of algebraic formulae with proofs and understandable examples to learn how to use them mathematically.

Binomial identities

The list of standard algebraic identities to expand the binomials, which have exponents.

$(1). \,\,\,$ $(a+b)^2$ $\,=\,$ $a^2+b^2+2ab$

$(2). \,\,\,$ $(a-b)^2$ $\,=\,$ $a^2+b^2-2ab$

$(3). \,\,\,$ $(a+b)(a-b)$ $\,=\,$ $a^2-b^2$

$(4). \,\,\,$ $(a+b)^3$ $\,=\,$ $a^3+b^3+3ab(a+b)$

$(5). \,\,\,$ $(a-b)^3$ $\,=\,$ $a^3-b^3-3ab(a-b)$

Special Binomials Products

The list of standard algebraic identities to multiply some special form binomials.

$(1). \,\,\,$ $(x+a)(x+b)$ $\,=\,$ $x^2+(a+b)x+ab$

$(2). \,\,\,$ $(x+a)(x-b)$ $\,=\,$ $x^2+(a-b)x-ab$

$(3). \,\,\,$ $(x-a)(x+b)$ $\,=\,$ $x^2-(a-b)x-ab$

$(4). \,\,\,$ $(x-a)(x-b)$ $\,=\,$ $x^2-(a+b)x+ab$

Factor form identities

There are two types of algebraic identities to factorize (or factorise) the algebraic expressions.

$(1).\,\,$ $a^2-b^2$ $\,=\,$ $(a+b)(a-b)$

$(2).\,\,$ $x^2-y^2$ $\,=\,$ $(x+y)(x-y)$

$(3).\,\,$ $a^3-b^3$ $\,=\,$ $(a-b)(a^2+b^2+ab)$

$(4).\,\,$ $x^3-y^3$ $\,=\,$ $(x-y)(x^2+y^2+xy)$

Trinomial identities

The list of standard algebraic identities to expand the squares of trinomials.

$(1). \,\,\,$ ${(a+b+c)}^2$ $=$ $a^2$ $+$ $b^2$ $+$ $c^2$ $+$ $2ab$ $+$ $2bc$ $+$ $2ca$

$(2). \,\,\,$ ${(a+b-c)}^2$ $=$ $a^2$ $+$ $b^2$ $+$ $c^2$ $+$ $2ab$ $-$ $2bc$ $-$ $2ca$

$(3). \,\,\,$ ${(a-b+c)}^2$ $=$ $a^2$ $+$ $b^2$ $+$ $c^2$ $-$ $2ab$ $-$ $2bc$ $+$ $2ca$

$(4). \,\,\,$ ${(a-b-c)}^2$ $=$ $a^2$ $+$ $b^2$ $+$ $c^2$ $-$ $2ab$ $+$ $2bc$ $-$ $2ca$

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