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The 26th degree/FOIL Frenzy 

Problem Statement
What is (x-a)(x-b)(x-c)...(x-y)(x-z)? 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 (x-a) (x-b).  Our teacher showed us how that makes the “happy martian”, using FOIL.  Then I multiplied that answer, x 2  -bx –ax +ab, times (x-c) and got x 3  - bx 2  - ax 2 + abxcx 2  + bcx +acxabc.   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 (x-x), and x-x 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 short-cutted this problem by first writing out every binomial that would be multiplied together.  By doing that, I would have seen (x-x) 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

Link(s) to resources, references, lesson plans, and/or other materials




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