Intermath | Workshop Support

 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

In this problem I will be multiplying 26 binomials together to determine the product.

Plans to Solve/Investigate the Problem

I will begin by multiplying the first 2 binomials together.  Then I will multiply the result by the 3rd binomial and so on until I have multiplied all 26 binomials together.

Investigation/Exploration of the Problem

I began by multiplying the first 2 binomials together using the FOIL method. I noticed that the product produced had 4 terms.  After multiplying by the 3rd binomial, I noticed that the product produced 8 terms.  Upon further exploration I began seeing a pattern emerge between the number of binomial factors and powers of 2.  I determined that the final product would contain 226, or 67,108,864 terms.  This did not appear to be a feasible task.

Upon further exploration I discovered that the 24th binomial was (x-x).  What a break!  (x-x) = 0.

Thanks to the zero property of multiplication my problem was solved.  Zero times anything is zero.

Extensions of the Problem

If 226 is equal to 67,108,864, what is 227?  The final answer is 134,217,728.

Author & Contact
Insert name and contact information.
lparsons@rockdale.k12.ga.us

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