Difference between revisions of "Simon's Favorite Factoring Trick"

Revision as of 16:04, 10 November 2007

Introduction

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo. <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced.

Statement of the factorization

The general statement of SFFT is: $\displaystyle {xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$ . Two special cases appear most commonly: $\displaystyle xy + x + y + 1 = (x+1)(y+1)$ and $\displaystyle xy - x - y +1 = (x-1)(y-1)$ .

Applications

This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually $\displaystyle {x}$ and $\displaystyle {y}$ are variables and $\displaystyle j,k$ are known constants. Also it is typically necessary to add the $\displaystyle {j}{k}$ term to both sides to perform the factorization.

Examples

AIME 1987/5

2000 AMC 12/Problem 6

@@ Line 1: / Line 1: @@
 == Introduction ==
-'''Simon's Favorite Factoring Trick''' (abbreviated SFFT) is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]].  [http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=factoring&t=8215 This] appears to be the thread where Simon's favorite factoring trick was first introduced.
+'''Simon's Favorite Factoring Trick''' (abbreviated SFFT) is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]].  <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced.
 == Statement of the factorization ==

Page

Toolbox

Search