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

Revision as of 19:01, 25 December 2007

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo. <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced.

Statement of the factorization

The general statement of SFFT is: ${xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$ . Two special cases appear most commonly: $xy + x + y + 1 = (x+1)(y+1)$ and $xy - x - y +1 = (x-1)(y-1)$ .

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.

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)

Olympiad

This problem has not been edited in. If you know this problem, please help us out by adding it.

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-== Introduction ==
 '''Simon's Favorite Factoring Trick''' (abbreviated SFFT) is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]].  <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced.
 == Statement of the factorization ==
-The general statement of SFFT is: <math>\displaystyle {xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>.  Two special cases appear most commonly: <math>\displaystyle xy + x + y + 1 = (x+1)(y+1)</math> and <math>\displaystyle xy - x - y +1 = (x-1)(y-1)</math>.
+The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>.  Two special cases appear most commonly: <math>xy + x + y + 1 = (x+1)(y+1)</math> and <math>xy - x - y +1 = (x-1)(y-1)</math>.
 == Applications ==
-This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>\displaystyle {x}</math> and <math>\displaystyle {y}</math> are variables and <math>\displaystyle j,k</math> are known constants. Also it is typically necessary to add the <math>\displaystyle {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.
 == Problems ==

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