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### Factoring Higher Order Polynomials

by Tim Isbell,  October 2015

Algebra 2 is a difficult subject for most high school students. In this class, factoring, graphing, and otherwise working with polynomial functions is an important topic. By "higher order" I mean functions like this one (graph shown):

• f(x) = x5 - 3x4 + 5x3 - x2 - 6x + 4

One Wednesday night this year, near the end of our Math Coaching session, I was coaching Algebra 1 when another coach and his Algebra 2 student asked for help factoring a cubic equation. I remembered factoring these before but quickly realized that it would take me 30-60 minutes to refresh my understanding of the relevant theorems and tricks from past years.

So over the next couple of days I sorted out the issues, applied them to this student's problem, and put it into a Google Doc to send her. Since we face this same math problem every year, I polished the Google Doc into a crib sheet for my future use, for other coaches, and for our students. To make it easy to access, I'm posting it here on this site.  The Google Doc crib sheet is embedded below. You can also access and download the Google Doc file here

#### Factoring higher order polynomials

A few weeks later, while working with another student, I ran across another interesting example. This one is a quartic, which means it is a fourth order function. We solved this problem using the same strategy as above, but it became a little tricky because it turned out to have all four roots at the same point! After solving it the long way, I realized that we could have also used the Binomial Theorem - but the Binomial Theorem was not yet in our students' toolbox. So I decided to write this one up, too, because it is a good snapshot of the intersection between factoring a conventional high ordered polynomial and the special case of a perfect quartic (or higher level perfect polynomial. The example is embedded below, and  you can access the underlying document here

#### Factoring a perfect quartic polynomial

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