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: x^4+4

[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 4x^2 to make the expression a perfect square trinomial.

x^4+4x^2-4,

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.

x^4+4x^2+4-4x^2, we add and subtract 4x^2 or like adding zero.

(x^2+2)^2-(2x)^2 observe that this is difference of two squares in the form of u^2-v^2=(u+v)(u-v) where u=x^2+2 and v=2x

(x^2+2+2x)(x^2+2-2x)

Rearranging the expression accordingly,

(x^2+2x+2)(x^2-2x+2) [/toggle]

 

Sample Problem 2:

Find all positive integral pairs (x,y) that satisfy the equation xy+x-2y=9.

[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.

xy+x-2y=9 xy+x-2y-2=9-2 x(y+1)-2(y+1)=7 (y+1)(x-2)=7

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

Case 1:

y+1=7   and   x-2=1

y=6       and   x=3

One pair is (3,6)

Case 2:

y+1=1   and   x-2=7

y=0       and   x=9

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

 

Therefore the only solution is (3,6) [/toggle]

 

 

 

 

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

Latest posts by Dan (see all)

You may also like...

5 Responses

  1. Deanne says:

    Hey very cool blog!! Man .. Beautiful .. Amazing .. I’ll bookmark your web site and take the feeds also…I am happy to find so many useful info here in the post, we need develop more strategies in this regard, thanks for sharing. . . . . .

  2. I have been exploring for a bit for any high quality articles or weblog posts on this kind of area . Exploring in Yahoo I finally stumbled upon this website. Reading this info So i’m satisfied to convey that I have an incredibly excellent uncanny feeling I came upon exactly what I needed. I so much without a doubt will make sure to do not omit this website and give it a look on a constant basis.

  3. Thank you of this blog. That’s all I’m able to say. You undoubtedly have made this web web site into an item thats attention opening in addition to critical. You certainly know a fantastic deal of about the niche, youve covered a multitude of bases. Fantastic stuff from this the main internet. All more than again, thank you for the blog.

  4. Youre so cool! I dont suppose Ive read anything in this way before. So good to uncover somebody with some original tips on this topic. realy appreciate starting this up. this excellent website is something that is necessary more than the internet, a person if we do originality. valuable function for bringing something new towards the web!

  5. Earline Dann says:

    Youre so cool! I dont suppose Ive read anything in this way before. So good to uncover somebody with some original tips on this topic. realy appreciate starting this up. this excellent website is something that is necessary more than the internet, a person if we do originality. valuable function for bringing something new towards the web!

Leave a Reply

Your email address will not be published. Required fields are marked *