Simon’s Favorite Factoring Trick

June 1, 2011 GB High School Algebra, Problem Solving and Proofs

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.

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Simon’s Favorite Factoring Trick

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