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Write-up


Title
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

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.

 

(x-a)(x-b) = x² -bx –ax + ab

(x² -bx –ax + ab)(x-c) = x³ - bx² - ax² + abx - cx² + cbx + cax + cab

(x³ - bx² - ax² + abx - cx² + cbx + cax + cab)(x-d) = ………

 

After further discussion, we realized that using the given pattern in the problem, (x-x) would eventually come up.  Because any number minus that number equals 0, we know that (x-x) 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 (x-x) 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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