Simon's Favorite Factoring Trick
Introduction
Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user user:ComplexZeta, whose name is Simon. This appears to be the thread where Simon's favorite factoring trick was first introduced.
Statement of the factorization
The general statement of SFFT is: . More oftenly however SFFT is introduced as
or
.
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
. This makes things much simpler. The rest of the problem is left as an exercise to the reader.