The 26th Degree – Foil Frenzy
What is (x-a)(x-b)(x-c)...(x-y)(x-z)? Explain how you found the
To solve this problem I will
need to find the product of 26 binomials.
Solve/Investigate the Problem
I will begin by multiplying the
first two binomials, recording an answer and multiplying by the next
binomial, recording an answer. I will continue with this pattern and look
for a pattern to short cut to the solution.
of the Problem
(x-a)(x-b) = x² - ax – bx + ab
(x² - ax – bx
+ ab)(x-c) = x³ - ax² - bx² - cx²
+ abx + acx + bcx –abc
(x³ - ax² - bx² - cx²
+ abx + acx + bcx –abc)(x-d) = x- ax³ - bx³ -cx³ -dx³ + abx² +
acx² + bcx²+ adx² + bdx²
– acdx – bcdx
– abcx +abcd
At this point I could see
patterns developing but could not see where I would have terms begin to
cancel one another out. I was frustrated at this point and did not continue
multiplying. Remember the instructor last week stated we would use
spreadsheets to solve lots of problems; I tried to create a spreadsheet
that might help. I was unable to use this route. Class discussion suggested
we use Pascal’s triangle. We agreed the first term in the solution
was x and the last
term would be +abcdefghijklmnopqrstuvwxyz. After
remembering what Pascal’s triangle was used for; we ruled out that avenue.
The class again went back to the fact that we needed some terms to cancel
out and when would this occur. Once we hit upon the fact that we needed to
involve x again, the term (x-x) immediately jumped out at me as zero. Using
the multiplication property of zero the solution to the problem must be 0.
Extensions of the Problem
Pascal’s triangle came up
as a possible method to solve the foil frenzy. Have students use Pascal’s
triangle to evaluate (x+a).
Author & Contact
Link(s) to vocabulary
Property of Zero