Euclidean Division of Polynomials: Theorem and Proof

First published at 06:58 UTC on September 23rd, 2017.

In this video I go over further into Euclidean Division and this time look at the theorem and algorithm for univariate (i.e. single-variable) polynomials. The theorem is very similar to that for integers which I covered in my earlier video, in that both cases require proving that the associated Euclidean Division Algorithm is valid to ultimately prove that the theorem is valid. The main difference between the two algorithms is that for the integer case each step involved adding 1 to the quotient; whereas for the polynomial case we need to apply a special value, s, to ensure that the degree of the remainder keeps decreasing incrementally.

The Euclidean Theorem for Division of Polynomials is that for any two univariate polynomials a and b, where b is not equal to 0 (to avoid the case of a/0 or dividing by zero) there are associated polynomials q and r called the quotient and remainder, respectively, such that a = bq + r and the degree of r is less than the degree of b. Note also that in case the remainder is 0 and the degree of b is 0, deg(0) is defined as negative. The Division Algorithm involves starting from q = 0 and then applying the theorem formula to obtain r and checking if the degree of r is less than the degree of b. If it is, then we are done. If it is not, then we add a special polynomial s, which I show is such that involves the leading coefficients of both the remainder and the divisor, b, as well as their degrees (or number of their highest power). The selection of s is such that the remainder keeps subtracting by 1 its degree until ultimately it is less than the degree of b. Thus ultimately the algorithm is proved and thus so is the theorem!

The theorem and associated algorithm is the basis for polynomial long division where a/b = q + r/b and can be rearranged so that theorem holds true a = bq + r. This is a truly fascinating concept and make sure to walk-through both the algorithm as well as several examples to make Euclidean Division as clear as possible, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh4EItHHqj75MMNeJbQ

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/euclidean-division-of-polynomials-theorem-and-proof

Related Videos:

Euclidean Division of Integers: Theorem and Proof: https://youtu.be/66juubotzi0
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
Long Division by Hand - An in depth look: http://youtu.be/giBZg5Vqryo
Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y .

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