Simon's Favorite Factoring Trick
Statement of the factorization
Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization. SFFT is: .
Credit
This factorization was first popularized by AoPS user ComplexZeta, whose name is Simon.
Applications
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually and
are variables and
are known constants. Also it is typically necessary to add the
term to both sides to perform the factorization.
Examples
(AIME 1987/5) and
are integers such that
. Find
.
Outline Solution: Rearrange to . The key step is changing the equation to
, where the equation factors to
, from which the problem is trivial to solve by applying some simple number theory.