# 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>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. | ||

− | == Problems == | + | == Amazing Practice Problems == |

− | ===Introductory=== | + | ===Introductory for Beginners=== |

*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? | *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? | ||

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([[2000 AMC 12/Problem 6|Source]]) | ([[2000 AMC 12/Problem 6|Source]]) | ||

− | ===Intermediate=== | + | ===Intermediate for Middle Class=== |

*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>. | *<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>. | ||

([[1987 AIME Problems/Problem 5|Source]]) | ([[1987 AIME Problems/Problem 5|Source]]) | ||

− | ===Olympiad=== | + | ===Olympiad for Elite=== |

*The integer <math>N</math> is positive. There are exactly 2005 ordered pairs <math>(x, y)</math> of positive integers satisfying: | *The integer <math>N</math> is positive. There are exactly 2005 ordered pairs <math>(x, y)</math> of positive integers satisfying: | ||

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Prove that <math>N</math> is a perfect square. (British Mathematical Olympiad Round 2, 2005) | Prove that <math>N</math> is a perfect square. (British Mathematical Olympiad Round 2, 2005) | ||

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== See Also == | == See Also == |

## Revision as of 00:27, 10 June 2019

## Contents

## About

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

## 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.

## Amazing Practice Problems

### Introductory for Beginners

- 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 for Middle Class

- are integers such that . Find .

(Source)

### Olympiad for Elite

- 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)