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

Revision as of 12:38, 23 June 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

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 == Statement of the factorization ==
-The general statement of SFFT is: <math>\displaystyle {xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>.  More oftenly however SFFT is introduced as <math>\displaystyle xy + x + y + 1 = (x+1)(y+1)</math> or <math>\displaystyle xy - x - y +1 = (x-1)(y-1)</math>.
+The general statement of SFFT is: <math>\displaystyle {xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>.  Two special cases appear most commonly: <math>\displaystyle xy + x + y + 1 = (x+1)(y+1)</math> and <math>\displaystyle xy - x - y +1 = (x-1)(y-1)</math>.
 == Applications ==

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