Title
The 26^{th} degree/FOIL Frenzy
Problem Statement
What is
(xa)(xb)(xc)...(xy)(xz)? Explain how you found the answer.
Problem setup
In this problem, I will have to
multiply 26 binomials together.
This could get really long and confusing; that’s my first
thought.
Plans to
Solve/Investigate the Problem
I know how to solve the
multiplication of 2 binomials, using FOIL, so I will probably multiply two of
them together, then multiply the next one times the first answer, etc. I’ll have to write each one
down and see if there are any patterns.
Investigation/Exploration
of the Problem
I started out by multiplying
(xa) (xb). Our teacher showed
us how that makes the “happy martian”,
using FOIL. Then I multiplied
that answer, x ^{2} bx –ax +ab, times (xc)
and got x ^{3 } bx ^{2 }^{
}ax ^{2} + abx – cx^{ 2} + bcx +acx – abc. To carry that out times 26
terms, it seems that the ending variable would be abcde….xyz. But I needed to see whether that
would be a + or a – abcde…xyz. In looking at the first couple
of answers, it appears that it
alternates between + and , with the even numbers having a + answer. Since there are 26 variables, and
that is an even number, then the final variable would be + also. As I was trying to fathom how large
the answer was, I realized that eventually (almost at the end of the
problem) I would come to something times (xx), and xx is equal to
zero. Anything times zero is
equal to zero, using the zero product property. Therefore, the answer has to be
zero. It was suggested that we could have shortcutted
this problem by first writing out every binomial that would be multiplied
together. By doing that, I
would have seen (xx) a lot sooner, and then realized quicker that the
answer had to be zero.
Extensions of the Problem
I have no idea, except to
perhaps always be on the lookout for tricky questions. In doing this problem, I did review
what a binomial, trinomial, and polynomial is. I learned about the “happy martian” picture made when doing FOIL. I reviewed a lot of the math I’d
forgotten, too.
Author & Contact
Meg Ramsey
mramsey@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
http://eduref.org/Virtual/Lessons/Mathematics/Algebra/ALG0002.html
http://www.intermathuga.gatech.edu/dictnary/descript.asp?termID=393
