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

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+ | ==About== | ||

'''Dr. Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]]. | '''Dr. Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]]. | ||

− | http://www.artofproblemsolving.com/community/c5h8215 | + | |

− | + | http://www.artofproblemsolving.com/community/c5h8215 appears to be the thread where Simon's favorite factoring trick was first introduced. | |

+ | ==The General Statement== | ||

+ | The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>. Two special common cases are: <math>xy + x + y + 1 = (x+1)(y+1)</math> and <math>xy - x - y +1 = (x-1)(y-1)</math>. | ||

The act of adding <math>{jk}</math> to <math>{xy}+{xk}+{yj}</math> in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square." | The act of adding <math>{jk}</math> to <math>{xy}+{xk}+{yj}</math> in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square." | ||

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== 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>jk</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>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>jk</math> term to both sides to perform the factorization. |

## Revision as of 03:20, 21 April 2015

## Contents

## About

**Dr. Simon's Favorite Factoring Trick** (abbreviated **SFFT**) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo.

http://www.artofproblemsolving.com/community/c5h8215 appears to be the thread where Simon's favorite factoring trick was first introduced.

## The General Statement

The general statement of SFFT is: . Two special common cases are: and .

The act of adding to in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."

## Applications

This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually and are variables and are known constants. Also, it is typically necessary to add the term to both sides to perform the factorization.

## Problems

### Introductory

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

(Source)

### Intermediate

- are integers such that . Find .

(Source)

### Olympiad

- The integer is positive. There are exactly 2005 ordered pairs of positive integers satisfying:

Prove that is a perfect square. (British Mathematical Olympiad Round 2, 2005)