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

Revision as of 19:17, 8 October 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 18: / Line 18: @@
 * [[Factoring]]
 * [http://www.artofproblemsolving.com/Movies/simon1.php AoPS's SFFT Video 1]
+*[http://www.artofproblemsolving.com/Movies/simon2.php AoPS's SFFT Video 2]
+*[http://www.artofproblemsolving.com/Movies/simon3.php AoPS's SFFT Video 3]

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