# Algebraic identities

An equation that expresses the equality property of two algebraic expressions is called an algebraic identity.

## Introduction

In algebra, an algebraic expression is equal to another algebraic expression in some cases. It means, the values of both algebraic expressions are equal. Hence, the equality property of both algebraic expressions is called an algebraic identity and it is used as a formula in mathematics. The following is the list of algebraic identities, which are used as formulas. So, learn each algebraic identity with proof to know how to use them as formulae in mathematics.

### 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.

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

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

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

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

### 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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