Difference between revisions of "Simon's Favorite Factoring Trick"
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'''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. | '''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 == | == Statement of the factorization == | ||
− | The general statement of SFFT is: <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 == | == Applications == | ||
− | This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math> | + | 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 == | == Problems == |
Revision as of 20:01, 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.
Contents
Statement of the factorization
The general statement of SFFT is: . Two special cases appear most commonly:
and
.
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.
Problems
Introductory
Two different prime numbers between and
are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
(Source)
Intermediate
are integers such that
. Find
.
(Source)
Olympiad
This problem has not been edited in. If you know this problem, please help us out by adding it.