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

To solve this problem I will need to find the product of 26 binomials.

Plans to Solve/Investigate the Problem

I will begin by multiplying the first two binomials, recording an answer and multiplying by the next binomial, recording an answer. I will continue with this pattern and look for a pattern to short cut to the solution.

Investigation/Exploration of the Problem

(x-a)(x-b) = x² - ax – bx + ab

(x² - ax – bx + ab)(x-c) = x³ - ax² - bx² - cx² + abx + acx + bcxabc

(x³ - ax² - bx² - cx² + abx + acx + bcxabc)(x-d) = x- ax³ - bx³ -cx³ -dx³ + abx² + acx² + bcx²+ adx² + bdx²

+cdx² -abdxacdxbcdxabcx +abcd

At this point I could see patterns developing but could not see where I would have terms begin to cancel one another out. I was frustrated at this point and did not continue multiplying. Remember the instructor last week stated we would use spreadsheets to solve lots of problems; I tried to create a spreadsheet that might help. I was unable to use this route. Class discussion suggested we use Pascal’s triangle. We agreed the first term in the solution was x and the last term would be +abcdefghijklmnopqrstuvwxyz. After remembering what Pascal’s triangle was used for; we ruled out that avenue. The class again went back to the fact that we needed some terms to cancel out and when would this occur. Once we hit upon the fact that we needed to involve x again, the term (x-x) immediately jumped out at me as zero. Using the multiplication property of zero the solution to the problem must be 0.

Extensions of the Problem

Pascal’s triangle came up as a possible method to solve the foil frenzy. Have students use Pascal’s triangle to evaluate (x+a).

Author & Contact
Dottie Mitcham

Foil Method

Binomial

Multiplication Property of Zero