Algebraic Division

During populace seek out flavor I'm afraid I can't resist looking at The Student Room - non to read all the infuriating pupil posts where they obsess most lost marks as well as degree boundaries, but only to come across each seek out newspaper every bit before long every bit possible. I receive got no patience. Arsey, presumably an anonymous teacher, posts model answers real before long later the exams receive got taken house (in previous years non until the adjacent day, but at nowadays at that topographic point are dissimilar papers for dissimilar fourth dimension zones the model answers are posted similar a shot later the U.K. sitting). When I was reading Arsey's recent C2 solutions, I noticed his method for polynomial division:

I've seen him do this before. It ever occurs to me that this is far to a greater extent than straightforward than the long partition method I instruct my students. Why do I instruct long division?  I suppose it's because it features inward the C2 textbook hence I've ever assumed it's the 'best' method. Perhaps adjacent yr I'll endeavor an alternative. In this shipping service I await at iv methods for polynomial division. 

1. Long Division

This method often features inward Influenza A virus subtype H5N1 flat textbooks. It exactly involves next a serial of steps (divide, multiply, subtract, pick out down, repeat) - an algorithm learnt past times drill rather than through understanding. The steps are familiar to those who learnt long partition at primary school. Those who were taught option partition methods (eg chunking) are at a slight disadvantage but do grab upwards quickly. Practice makes perfect. This isn't an elegant method - it's totally procedural as well as isn't peculiarly prissy to teach, as well as students receive got to know special rules for special cases (eg including a 0x term) - but it does the undertaking exactly fine. It works good when there's a remainder.

2. Grid/Box Method
I've exactly tried this method for the offset fourth dimension as well as I can't believe how tardily it is - as well as hence much quicker than long division! All y'all receive got to do is develop a multiplication grid - start past times filling inward the bits y'all know as well as hence the repose follows past times logic. This video from Bon Crowder explains the method real clearly. James Tanton calls this the Galley Method - his Curriculum Essay about how it works includes exercises as well as interesting questions. The motion painting below from Esther (@MrsMathematica) shows a sensible layout which makes dealing alongside remainders real easy.

3. Inspection
This is like the grid method but laid out differently. All y'all receive got to do is write your polynomial every bit the production of a linear business office as well as an unknown quadratic (or cubic, quartic etc, depending on the question) hence purpose logic as well as algebra to function out the numbers past times equating the coefficients. It's quick as well as fairly straightforward. It's also easier to follow what's going on than inward the confusing algorithm of long division.

In a 1940s textbook I spotted an alternative layout for inspection that I love. In the illustration below nosotros desire to dissever past times x - 1 hence nosotros write (x - 1) 3 times as well as hence exactly create total inward the rest. Again, it's quick as well as logical, as well as tardily to expire on rail of each term.

4. Synthetic Division

I don't similar this method so I don't truly desire to get upwards it here, but for completeness I suppose I should. I've works life lots of (often negative) reference to it on American websites but I've never seen it used inward England. I'm told that it's unremarkably used in Scotland (thanks to @mrallanmaths, @kenniejp23 as well as Paul Smith for commenting). The argue I dislike it is it appears to endure ane of those 'remember the steps but receive got no clue what's truly happening' methods.

Source: acedemic.utep.edu

I followed the rules inward a higher house as well as it did work, summation it was pretty quick, but it was a fighting similar doing a magic trick.

The animation below shows the equivalence of long partition as well as synthetic division. It looks to me that synthetic partition is exactly a confusing method made fifty-fifty to a greater extent than abstract. There are many defenders of synthetic partition though - they tell that it's an acceptable method providing students are taught the underlying concept earlier they start applying the super-efficient algorithm.

Source: purplemath.com

So that's it - iv methods for dividing polynomials. This PowerPoint from the Further Mathematics Support Programme summarises the offset 3 methods.

Which do y'all prefer?

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