Simon’s Favorite Factoring Trick

Hmmm... I didn't know Simon was that good in math.

When I was quite younger, one of my hobbies was joining internet forums (fora?) on problem solving. I was not really good at it, so my role was only to ask questions. One of the internet forums I joined was the Art of Problem Solving math forum.

Art of Problem Solving (AOPS) is a community of problem solvers dedicated for math competitions – probably the best place on the web to ask hard (and very hard)  math questions. One of the tricks I learned there was Simon’s Favorite Factoring Trick (SFFT), a factorization technique popularized by one AOPS member. The general strategy (see example 3)  of SFFT is to add a constant or variable to an expression to make it factorable. This strategy can also be named as “completing rectangle” in analogy with “completing the square.” 

Let’s have a few examples.

Example 1: Find all positive integer (x,y) such that xy + x + y = 20.

Solution: Using SFFT, we add 1 to both sides of the equation giving us xy + x + y + 1= 21. This gives us x(y+1) + (y+1) = 21 which is equivalent to (x+1)(y+1) = (7)(3). Therefore, x = 6 and y = 2.

Example 2: Find the length and the width of a rectangle whose area is equal to its perimeter.

Solution: Let a and b be the length and width of the rectangle. Since its perimeter is equal to its area, it follows that 2(a + b) = ab, which is equivalent to ab - 2a - 2b = 0.  Adding 4 to both sides of the equation, we have ab - 2a - 2b + 4 = 4.  Factoring the left hand side, we have a(b-2) - 2(b-2) = 4. This gives us (a-2)(b-2) = 4. Therefore a = 4 and b=4, or a = 6 and b = 3.  The latter satisfies the condition above, so a = 6 and b = 3.

Example 3: SFFT can be used in general for equations of the form xy + ax + by = 0.  This simplify to (y + a) + by + ba = ba. This simplifies to x(y + a) + b(y+a) = ba which is equal to (y + a)(x+b) = ba.

Art of Problem Solving is good place for gifted math students. I bought two of their books a year ago, The Art of Problem Solving: The Basics and  The Art of Problem Solving: And Beyond, and until now, I have not yet finished solving all the problems. They are excellent books if you are preparing for math competitions.

Photo Credit: Wikipedia

 

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