# Simon’s Favorite Factoring Trick

Another elegant factoring technique that will solve hard problems is Simon’s Favorite Factoring Trick or commonly abbreviated as SFFT. This technique was popularized by Simon Rubinstein-Salzedo, a user of Art of Problem Solving. The idea is adding a term to make the whole expression factorable. Try to do the problem first by yourself before looking at the solution for you to evaluate your skill and maximize your learning.

** **

**Sample Problem 1:**

Factor completely:

[toggle title=”Solution:”]

Observe that the two terms are perfect squares. So we can add a term that can make the whole expression a perfect square trinomial.

We can add to make the expression a perfect square trinomial.

,

However, in the rule of math, we do cannot alter the equation or expression to keep its original form. However adding a zero to a whole expression will not change anything on the original expression.

, we add and subtract or like adding zero.

observe that this is difference of two squares in the form of where and

Rearranging the expression accordingly,

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**Sample Problem 2:**

Find all positive integral pairs that satisfy the equation .

[toggle title=”Solution:”]

Look at the left hand side of equation. Think of a number that you can add or subtract to make it factorable. In this equation we subtract the left side by 2.

7 is a prime number which means that the factor of 7 is only 1 and itself. By trial and error,

Case 1:

and

and

One pair is

Case 2:

and

and

Since we are ask for positive integral pairs this is not a solution since y=0.

Therefore the only solution is [/toggle]

### Dan

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