The twenty-sixth degree/Foil frenzy
(x-a)(x-b)(x-c)...(x-y)(x-z)? Explain how you found the answer.
We were asked to solve the
above problem which seemed like a normal polynomial problem requiring the
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.
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 “abcdef…yz”. 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
be positive because of the exponential value being positive.
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