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

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== Statement of the factorization ==
 
== Statement of the factorization ==
The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>.  More oftenly however SFFT is introduced as <math>xy + x + y + 1 = (x+1)(y+1)</math> or <math> xy - x - y +1 = (x-1)(y-1)</math>.   
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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>.   
  
 
== Applications ==
 
== Applications ==
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>{x}</math> and <math>{y}</math> are variables and <math>j,k</math> are known constants. Also it is typically necessary to add the <math>{j}{k}</math> term to both sides to perform the factorization.
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This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>\displaystyle {x}</math> and <math>\displaystyle {y}</math> are variables and <math>\displaystyle j,k</math> are known constants. Also it is typically necessary to add the <math>\displaystyle {j}{k}</math> term to both sides to perform the factorization.
  
 
== Examples ==
 
== Examples ==
  
 
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=454191#p454191 AIME 1987/5]
 
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=454191#p454191 AIME 1987/5]

Revision as of 22:18, 21 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: $\displaystyle {xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$. More oftenly however SFFT is introduced as $\displaystyle xy + x + y + 1 = (x+1)(y+1)$ or $\displaystyle xy - x - y +1 = (x-1)(y-1)$.

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

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