Simon's Favorite Factoring Trick

Revision as of 22:10, 28 May 2018 by Music art (talk | contribs) (The General Statement)

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: ${xy}+{xk}+{jy}+{jk}=(x+j)(y+k)$. Two special common cases are: $xy + x + y + 1 = (x+1)(y+1)$ and $xy - x - y +1 = (x-1)(y-1)$.

The act of adding ${jk}$ to ${xy}+{xk}+{jy}$ in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square." n

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 $jk$ term to both sides to perform the factorization.

Problems

Introductory

  • Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$\mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$

(Source)

Intermediate

  • $m, n$ are integers such that $m^2 + 3m^2n^2 = 30n^2 + 517$. Find $3m^2n^2$.

(Source)

  • The integer $N$ is positive. There are exactly $2005$ pairs $(x, y)$ of positive integers satisfying:

\[\frac 1x +\frac 1y = \frac 1N\]

Prove that $N$ is a perfect square. (British Mathematical Olympiad Round 2, 2005)

See Also

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