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Simon's Favorite Factoring Trick

Revision as of 20:53, 17 June 2006 by Chess64 (talk | contribs) (→‎Applications)

Statement of the factorization

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization. SFFT is: ${xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$.

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 ${x}$ and ${y}$ are variables and $j,k$ are known constants. Also it is typically necessary to add the ${j}{k}$ term to both sides to perform the factorization.

Examples

(AIME 1987/5) $m$ and $n$ are integers such that $m^2 + 3m^2n^2 = 30n^2 + 517$. Find $3m^2n^2$.

Outline Solution: Rearrange to $m^2 + 3m^2n^2 -30n^2= 517$. The key step is changing the equation to $m^2 + 3m^2n^2 -30n^2-10= 507$, where the equation factors to $(3n^2 + 1)(m^2 - 10) = 507 = 3\cdot 13^2$, from which the problem is trivial to solve by applying some simple number theory.

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