Intermath | Workshop Support


The 26th Degree (The exciting world of FOIL Frenzy)

Problem Statement
What is (x-a)(x-b)(x-c)...(x-y)(x-z)? Explain how you found the answer.

Problem setup

This will require that twenty-six 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 (x-a)(x-b).  Then multiply the answer to the next binomial (x-c).  Then multiply the answer to the next binomial (x-d).  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=22, 8 terms = 23, 16 terms = 24,… Therefore, multiplying (x-z) would be term 226.  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 26th 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 24th binomial (x-x) would be 0 which in turn when multiplied with the previous answer would yield a product of 0.



Extensions of the Problem

Substitute twenty-six consecutive numbers in place of the alphabets and solve for the value of x.

Author & Contact

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