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

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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>{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>{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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+ | === Examples === | ||

+ | ([[AIME]] 1987/5) <math>m</math> and <math>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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+ | '''Outline Solution:''' Rearrange to <math>m^2 + 3m^2n^2 -30n^2= 517</math>. The key step is changing the equation to <math>m^2 + 3m^2n^2 -30n^2-10= 507</math>, where the equation factors to <math>(3n^2 + 1)(m^2 - 10) = 507 = 3\cdot 13^2</math>, from which the problem is trivial to solve by applying some simple number theory. |

## Revision as of 21:53, 17 June 2006

### Statement of the factorization

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization. SFFT is: .

### Credit

This factorization was first popularized by AoPS user ComplexZeta, whose name is Simon.

### 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.

### Examples

(AIME 1987/5) and are integers such that . Find .

**Outline Solution:** Rearrange to . The key step is changing the equation to , where the equation factors to , from which the problem is trivial to solve by applying some simple number theory.