An algebraic identity is an equality that holds for any values of its variables.
For example, the identity holds for all values of and .
Since an identity holds for all values of its variables, it is possible to substitute instances of one side of the equality with the other side of the equality. For example, because of the identity above, we can replace any instance of with and vice versa.
Clever use of identities offers shortcuts to many problems by making the algebra easier to manipulate. Below are lists of some common algebraic identities.
The following identities are product formulas that are examples of the binomial theorem:
The following identities are factoring formulas; more general factoring formulas are listed in the wiki Algebraic Manipulation - Identities:
The following identities can be derived by some clever factoring and manipulation of the terms:
The identity holds for all real values of . What is
Multiplying out the left side of the identity, we have This expression must be equal to the right hand side of the identity, implying so , , and , and .
If and , what is
While it is possible to solve for and , a more elegant solution exploits the identity which can be rewritten as Substituting in and for and we get
The following identity holds for all real numbers and What is
Rewriting the identity, we have This gives the following system of equations: Solving this system of equations gives
For all real numbers and such that the following identity holds: What is
Since implies substituting this into the given identity gives Since this is an identity in we have
If and what is
Since holds for all real values of and we have
If and what is
Substituting and into the identity we have