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Why Are Remainders Small?

Date: 07/06/2016 at 22:18:47
From: Chris
Subject: degree of remainder less than degree of divisor

Dear Dr. Math, 

Polynomial divisions always seem to show that the degree of the remainder
is less than that of the divisor. Is there a simple explanation for why 
this is so?

I mean, we can say that the degree of P(x) is greater than the degree of
D(x) in

   P(x)/D(x) = Q(x) + R(x)/D(x)
   
But how can I conclude this?

   degree of R(x) < degree of D(x)

Could you please explain why the degree of the remainder should be less
than the degree of the divisor? 

Thank you for your time and help.



Date: 07/06/2016 at 23:33:59
From: Doctor Peterson
Subject: Re: degree of remainder less than degree of divisor

Hi, Chris.

It's not something you can conclude -- but it is something you can
require.

That is, the "division algorithm" is a theorem that says you can always
find Q and R such that this will be true:

   P(x) = Q(x)D(x) + R(x)]

They will be unique if we require that the degree of R be less than that
of D. In essence, we are DEFINING the remainder as the polynomial that
fits in that equation such that its degree is less than that of the
divisor, and proving that this always exists and is unique.

We have an analogous situation in the division of integers. Given numbers
p and d, we can find MANY numbers q and r such that

   p = qd + r

We define the quotient and remainder uniquely by requiring that r < d. For
example, given p = 7 and d = 3, we can write ...

   7 = 0*3 + 7
   7 = 1*3 + 4
   7 = 2*3 + 1
   7 = 3*3 - 2

... and so on. But only one of these has 0 <= r < d, namely

   7 = 2*3 + 1

So we say that the quotient of 7/3 is 2, and the remainder is 1. 

It's not that it has to be less than 3 to write the equation, but rather
that we WANT it to be less than 3 so that there is only one valid answer.

To put it another way, if the "remainder" you get is NOT less than 3, you
can always increase the quotient so that it will be. And going back to
polynomials, if the degree of R is not less than that of D, you can always
go another step and bring the degree down to where you want it.

Does that help?

- Doctor Peterson, The Math Forum at NCTM
  http://mathforum.org/dr.math/ 



Date: 07/07/2016 at 00:20:44
From: Chris
Subject: Thank you (degree of remainder less than degree of divisor)

Dear Doctor Peterson, 

Thank you so much for your prompt reply and detailed answers!!!

I wasn't able to find satisfactory answers elsewhere on the Internet. But
your explanation is very clear, and helps me a great deal.

Again, thank you!!!

Sincerely,
Chris
Associated Topics:
High School Polynomials
Middle School Division

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