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

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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.
 
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 ==
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== Problems ==
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===Introductory===
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*
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Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
  
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=454191#p454191 AIME 1987/5]
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<math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }  </math>
  
*[[2000 AMC 12/Problem 6]]
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([[2000 AMC 12/Problem 6|Source]])
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===Intermediate===
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*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>.
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([[1987 AIME Problems/Problem 5|Source]])
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===Olympiad===
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{{problem}}
  
 
== See Also ==
 
== See Also ==

Revision as of 17:10, 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.

Problems

Introductory

Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$\mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$

(Source)

Intermediate

  • $m, n$ are integers such that $m^2 + 3m^2n^2 = 30n^2 + 517$. Find $3m^2n^2$.

(Source)

Olympiad

This problem has not been edited in. If you know this problem, please help us out by adding it.

See Also

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