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 20:53, 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.
