Simon's Favorite Factoring Trick

Introduction

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo. This appears to be the thread where Simon's favorite factoring trick was first introduced.

Statement of the factorization

The general statement of SFFT is: ${xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$ . More oftenly however SFFT is introduced as $xy + x + y + 1 = (x+1)(y+1)$ or $xy - x - y +1 = (x-1)(y-1)$ .

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 $(3n^2 + 1)(m^2 - 10) = 507 = 3\cdot 13^2$ . This makes things much simpler. The rest of the problem is left as an exercise to the reader.

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Simon's Favorite Factoring Trick

Contents

Introduction

Statement of the factorization

Applications

Examples