An algebraic identity is an equality that remains true for every value of its variables for which both sides are defined. For example, (a+b)² = a²+2ab+b². An equation such as x+3=8 is different: it is true only for a particular value of x. To check an identity, expand or factor one side using valid algebraic steps, then compare it with the other side.

The identities (a+b)²=a²+2ab+b² and (a−b)²=a²−2ab+b² can be understood by dividing a square into smaller regions. The two cross-regions each have area ab, which explains the middle term 2ab. These identities let us expand expressions such as (3x+2)² without multiplying every term from scratch.

Multiplying conjugate binomials gives (a+b)(a−b)=a²−b² because the +ab and −ab terms cancel. This difference-of-squares identity works in both directions: it expands a product and factors an expression such as 25x²−9=(5x−3)(5x+3). Always check that both terms are squares before applying it.

Factorisation rewrites a sum or difference as a product. First remove any common factor. Then check for a familiar identity, such as a²−b² or a²±2ab+b². For a quadratic trinomial, look for two binomials whose product reproduces every term. For example, 2x²+7x+3=(2x+1)(x+3), as expansion verifies.

A rational expression is a quotient of polynomials. Its denominator must not be zero. For example, (x²−9)/(x−3)=(x−3)(x+3)/(x−3)=x+3 only when x≠3. Cancelling a common factor simplifies the expression but does not restore excluded values to its domain. State these restrictions before simplifying.