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

m (About)
m (Problems)
Line 12: Line 12:
 
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.
  
== Problems ==
+
== Amazing Practice Problems ==
===Introductory===
+
===Introductory for Beginners===
 
*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?
  
Line 19: Line 19:
  
 
([[2000 AMC 12/Problem 6|Source]])
 
([[2000 AMC 12/Problem 6|Source]])
===Intermediate===
+
===Intermediate for Middle Class===
 
*<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 for Elite===
  
 
*The integer <math>N</math> is positive. There are exactly 2005 ordered pairs <math>(x, y)</math> of positive integers satisfying:
 
*The integer <math>N</math> is positive. There are exactly 2005 ordered pairs <math>(x, y)</math> of positive integers satisfying:
Line 30: Line 30:
  
 
Prove that <math>N</math> is a perfect square. (British Mathematical Olympiad Round 2, 2005)
 
Prove that <math>N</math> is a perfect square. (British Mathematical Olympiad Round 2, 2005)
 
  
 
== See Also ==
 
== See Also ==

Revision as of 00:27, 10 June 2019

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: ${xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$. 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 ${xy}+{xk}+{yj}$ in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."


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.

Amazing Practice Problems

Introductory for Beginners

  • 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 for Middle Class

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

(Source)

Olympiad for Elite

  • 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. (British Mathematical Olympiad Round 2, 2005)

See Also

Invalid username
Login to AoPS