Factorization is the decomposition of an expression into a product of its factors.
The following are common factorizations.
For any positive integer , In particular, for , we have .
For an odd positive integer ,
Factorization often transforms an expression into a form that is more easily manipulated algebraically, has easily recognizable solutions, and gives rise to clearly defined relationships.
Find all ordered pairs of integer solutions such that .
We have . Since the factors and on the right hand side are integers whose product is a power of 2, both and must be powers of 2. Furthermore, their difference is
implying the factors must be and . This gives , and thus . Therefore, is the only solution.
Factorize the polynomial
Observe that if , then ; if , then ; and if , then . By the Remainder-Factor Theorem, and are factors of . This allows us to factorize