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

Revision as of 15:05, 17 July 2006

Introduction

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo. 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: $\displaystyle {xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$ . 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 11: / Line 11: @@
 * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=454191#p454191 AIME 1987/5]
+*[[2000 AMC 12/Problem 6]]
 == See Also ==

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