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Assignment #1:  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 am going to have to multiply 26 binomials together. 


Plans to Solve/Investigate the Problem

My initial plan was to multiply each binomial one by one, using the distributive property, until I reached the last term in the sequence. 


Investigation/Exploration of the Problem

For the first terms in the problem, I utilized the mnemonic, FOIL, to distribute my terms through the binomial.  After figuring that multiplying each binomial individually was a tedious process, I looked for short cuts to solving the problem.  I did notice a pattern of multiplying the binomials, which helped me to determine the first and last term in the sequence.  The first term in the sequence was always x, followed by an exponent representative of its place in the alphabet.  So, after using the distributive property the first time, the first term was x².  After performing the distributive property for the third time, the term was x³, and so on.  I used my knowledge of positive and negative integers to determine that the last term in the sequence would be positive.  I also noticed that the last term in the sequence would be the product of all the letters together.  For example, abcdef…  As a group, we created a chart that showed the patterns of terms after multiplying each successive binomial.  Multiplying the first two binomials would produce 4 terms or 2².  Multiplying the next binomial would produce 8 terms or 2³, and so on.  This process would have produced 2terms, which would have been too large to determine.  So, I had to look for an alternative to solve the problem.  After much thought, I determined that one of the binomials in the sequence was (x-x), which results in an answer of 0.  According to the multiplicative property of zero, anything multiplied by 0 is 0.  Thus, the answer is 0. 


Extensions of the Problem

There were many extensions in this problem, which were mentioned in my investigation.  Other areas discussed include quadrants, coordinate system, and quadratic equations.  More specifically, we talked about the four quadrants created by the coordinate plane and how it may relate to the naming of the quadratic equation.  This led into a discussion of parabolas, and how to graph solutions to a set. 

Author & Contact
Mesha Rainey

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

Distributive Property

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