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

(The General Statement)
m (The General Statement)
(63 intermediate revisions by 43 users not shown)
Line 1: Line 1:
  
==About==
+
==The General Statement==
'''Dr. Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]].
+
Simon's Favorite Factoring Trick (SFFT) is often used in a Diophantine(Positive) equation where factoring is needed. The most common form it appears is when there is a constant on one side of the equation and a product of variables with each of those variables in a linear term on the other side. A extortive example would be: <cmath>xy+66x-88y=23333</cmath>where <math>23333</math> is the constant term, <math>xy</math> is the product of the variables, <math>66x</math> and <math>-88y</math> are the variables in linear terms.
 +
 
 +
 
 +
Let's put it in general terms. We have an equation <math>xy+jx+ky=a</math>, where <math>j</math>, <math>k</math>, and <math>a</math> are integral constants. According to Simon's Favourite Factoring Trick, this equation can be transformed into: <cmath>(x+k)(y+j)=a+jk</cmath>
 +
Using the previous example, <math>xy+66x-88y=23333</math> is the same as: <cmath>(x-88)(y+66)=(23333)+(-88)(66)</cmath>
  
==The General Statement==
 
The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>.  Two special common cases are: <math>xy + x + y + 1 = (x+1)(y+1)</math> and <math>xy - x - y +1 = (x-1)(y-1)</math>.
 
  
The act of adding <math>{jk}</math> to <math>{xy}+{xk}+{yj}</math> in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."
+
If this is confusing or you would like to know the thought process behind SFFT, see this eight-minute video by Richard Rusczyk from AoPS: https://www.youtube.com/watch?v=0nN3H7w2LnI. For the thought process, start from https://youtu.be/0nN3H7w2LnI?t=366
  
 
== Applications ==
 
== Applications ==
 
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>jk</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>jk</math> term to both sides to perform the factorization.
BID DJD<DlkDJDHDdcc
 
  
== Problems ==
+
== Fun Practice Problems ==
 
===Introductory===
 
===Introductory===
 
*Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
 
*Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
  
<math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }  </math>
+
<math> \mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }  </math>
  
 
([[2000 AMC 12/Problem 6|Source]])
 
([[2000 AMC 12/Problem 6|Source]])
 +
 
===Intermediate===
 
===Intermediate===
 
*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>.
 
*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>.
  
 
([[1987 AIME Problems/Problem 5|Source]])
 
([[1987 AIME Problems/Problem 5|Source]])
 +
 
===Olympiad===
 
===Olympiad===
  
Line 29: Line 32:
 
<cmath>\frac 1x +\frac 1y = \frac 1N</cmath>
 
<cmath>\frac 1x +\frac 1y = \frac 1N</cmath>
  
Prove that <math>N</math> is a perfect square. (British Mathematical Olympiad Round 2, 2005)
+
Prove that <math>N</math> is a perfect square.  
 
 
 
 
 
 
 
 
 
 
 
 
  
Friend DJ835689
+
Source: (British Mathematical Olympiad Round 3, 2005)
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
Friend DJ835689
 
  
== See Also ==
+
== See More==
 
* [[Algebra]]
 
* [[Algebra]]
 
* [[Factoring]]
 
* [[Factoring]]

Revision as of 20:18, 12 January 2021

The General Statement

Simon's Favorite Factoring Trick (SFFT) is often used in a Diophantine(Positive) equation where factoring is needed. The most common form it appears is when there is a constant on one side of the equation and a product of variables with each of those variables in a linear term on the other side. A extortive example would be: \[xy+66x-88y=23333\]where $23333$ is the constant term, $xy$ is the product of the variables, $66x$ and $-88y$ are the variables in linear terms.


Let's put it in general terms. We have an equation $xy+jx+ky=a$, where $j$, $k$, and $a$ are integral constants. According to Simon's Favourite Factoring Trick, this equation can be transformed into: \[(x+k)(y+j)=a+jk\] Using the previous example, $xy+66x-88y=23333$ is the same as: \[(x-88)(y+66)=(23333)+(-88)(66)\]


If this is confusing or you would like to know the thought process behind SFFT, see this eight-minute video by Richard Rusczyk from AoPS: https://www.youtube.com/watch?v=0nN3H7w2LnI. For the thought process, start from https://youtu.be/0nN3H7w2LnI?t=366

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.

Fun Practice 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?

$\mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$

(Source)

Intermediate

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

(Source)

Olympiad

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

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

Prove that $N$ is a perfect square.

Source: (British Mathematical Olympiad Round 3, 2005)

See More

Invalid username
Login to AoPS