Title
The 26^{th} Degree / FOIL Frenzy
Problem Statement
What is
(xa)(xb)(xc)...(xy)(xz)? Explain how you found the answer.
Problem setup
Restate the problem in your own
words. Discuss mathematical similarities between this problem and other
problems you have solved.
In this problem, you are asked
to multiply 26 binomials and figure out what the final answer is.
Plans to
Solve/Investigate the Problem
Discuss your initial
plans/strategies/technologies toward the solution of the problem.
My initial plan was to start
with the first two binomials and multiply them. After that I would continue to
multiply the product of the first two binomials times the third and so
on and so forth. I hoped to
find a pattern that would form that would allow me to figure out what the
final equation would be.
Investigation/Exploration
of the Problem
Carry out your plans/strategies
you planned initially. Give a well organized explanation and details about
how the problem was approached and explored. You should do that so
that the reader can follow/construct/understand your work with minimal
effort. Include numerical, graphical data (to the extend
that it is applicable) to support your arguments and conjectures. Try to
include multiple approaches/representations (numerical, graphical and
symbolic) to the problem and the solution. Label diagrams, tables, graphs,
or other visual representations you used. Provide an algebraic
proof/solution for your conjectures/observations where it's applicable.
(xa)(xb) = x² bx –ax + ab
(x² bx
–ax + ab)(xc) = x³  bx²  ax²
+ abx  cx² + cbx +
cax + cab
(x³  bx²  ax² +
abx  cx² + cbx + cax + cab)(xd) = ………
After further discussion, we
realized that using the given pattern in the problem, (xx) would eventually
come up. Because any number
minus that number equals 0, we know that (xx) would have to be zero. Because this is zero, the answer to
the question is zero. This is
because the zero property of multiplication states that any number
multiplied by zero is zero.
Extensions of the Problem
Discuss possible extensions for
the problem and explore/investigate at least one of the extensions you
discussed.
If (xx) never came up because
instead of using x as a constant in each binomial ψ was used to
represent the variable, how many terms would be in the final equation? To find the answer, we figured out
that we could raise two to the power of the number of letter in the
alphabet which is 26. The final
answer would have 2 which equals 67,108,864
terms in the final answer.
Author & Contact
Benjamin Moore
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