Title
The 26^{th} Degree (The exciting world of FOIL Frenzy)
Problem Statement
What is
(xa)(xb)(xc)...(xy)(xz)? Explain how you found the answer.
Problem setup
This will require that
twentysix binomials be multiplied together.
Plans to
Solve/Investigate the Problem
If actually carried out this
problem would require a lot of time, patience, and organizational
skills. The easiest and most
efficient way to attack this problem is to look for a potential pattern.
There are several mathematical
concepts and or topics that can be discussed for possible means to solving
this problem: distributive
property, polynomials, variables, exponents, quadrants, patterns)
Investigation/Exploration
of the Problem
It should be recognized that
the problem may involve the distributive property or the FOIL concept. Theses concepts involve multiplying
4 terms together (xa)(xb).
Then multiply the answer to the next binomial (xc). Then multiply the answer to the next
binomial (xd). From this point
there should be a noticeable pattern.
The number of terms doubled after each step (2, 4,8,16, 32,…).
Now these numbers can be changed into exponential form: 4 terms=2^{2}, 8 terms = 2^{3},
16 terms = 2^{4},… Therefore, multiplying (xz) would be term
2^{26}. This would be
67,108,864. The next problem to
solve would be to determine if this would be negative or positive. Within the pattern it can be noticed
that each odd number term (a, c, e,..) is negative and each even number
term (b, d, f, …) is positive.
Considering that z is the
26^{th} letter of the alphabet, this would be positive. But considering the zero property of multiplication
(anything times 0 will equal 0), this was an incorrect solution. The 24^{th} binomial (xx)
would be 0 which in turn when multiplied with the previous answer would
yield a product of 0.
Extensions of the Problem
Substitute twentysix
consecutive numbers in place of the alphabets and solve for the value of x.
Author & Contact
L.Laster
llaster@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
Link 1
Link 2
