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Title  
The twenty-sixth degree/Foil frenzy 

Problem Statement
What is (x-a)(x-b)(x-c)...(x-y)(x-z)? Explain how you found the answer.

Problem setup

We were asked to solve the above problem which seemed like a normal polynomial problem requiring the distributive property. 

 

Plans to Solve/Investigate the Problem

We began with a simple distribution of x to the variables, a and b using the FOIL method.  Foil of course quickly breaks down when using more than four terms. While we knew we had a finite number of variables (26), and we knew what they all were, it quickly became apparent that the number of variables compounded in exponential form was going to present a challenge. 

 

Investigation/Exploration of the Problem

We continued to add variables until we could determine that the first term would be “x to the twenty sixth power” and that the last one would be “abcdefyz”.  Someone mentioned Pascal’s triangle as a method of figuring out an answer more quickly.  I am unfamiliar with Pascal’s triangle so that didn’t help me.  We began to look for patterns and I looked for a way to combine terms in the hope that some formula would present itself.  I played the whole alphabet out through my head and realized that whatever pattern  that developed would eventually be thrown off by the reoccurrence of the variable “x”.  When you multiply “x” by itself, you come up with a value to the second power and that would mess up the pattern.  Then I realized that x – a, b, c, etc. would eventually arrive at “x-x” which is zero (!) and zero multiplied by the rest of our gigantic problem, no matter how large, will give the answer zero (using the zero product property).

 

Extensions of the Problem

As we multiplied the equations out, we could see that the number of terms in the answer would double after multiplying a new variable (ex.  (x-a)(x-b) ends up with four terms and when you multiply x-c to the equation your answer has eight terms in it).  We determined that the final answer IF x were not to come upon itself again (for example, if x were some other variable than a letter in the alphabet) would have 67,108,864 terms (whew!) and would take eons to complete.  Knowing that the doubling of terms takes place whenever a new term is added to the problem along with the fact that there are twenty-six letters in the alphabet, we were able to determine that the final answer would number “two to the twenty –sixth power” in terms, which is 67,108,864, as mentioned.  We also determined that the last term (abcdefghijyz) would be positive because of the exponential value being positive.  

 

Author & Contact
Kevin Smith
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Link(s) to resources, references, lesson plans, and/or other materials
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