1

I am trying to understand why polynomial long division algorithm works. I have found an answer on Quora, please follow the link: Why does polynomial long division work?

As you can see the explanation comes along with the example:

And the first step is calculation of the division of the leading terms of numerator and denominator and getting to know how much this differs from the original fraction. In this case:

What I can not understand is why we can perform this first step(why we can divide only leading terms of every polynomial). Could you please explain.

Thanks in advance.

user641597
  • 43
  • This is to replace the fraction $\dfrac{\text{dividend}}{\text{divisor}} with another fraction with a lower degree dividend, and so on, until the degree of the dividend ha lower degree than the divisor. – Bernard May 18 '19 at 16:46
  • It is so because the polynomial ring over the field $\mathbb{Q}$ is an Euclidean ring in which Euclid's division algorithm is valid. Read this with the comment of @Bernard – mark haokip May 18 '19 at 16:56
  • Key idea comparing lead terms in $,F = Q,G + R,$ shows that the lead term of the quotient $Q$ is the quotient of the lead terms of $F \ &\ G$ (the remainder $R$ doesn't contribute to the lead term because its degree is smaller, i.e. $,\deg R < \deg G \le \deg QG)\ \ $ – Bill Dubuque May 18 '19 at 17:01

0 Answers0