Negative Remainders in Arithmetic and Algebra: a Difference of Degrees

Date: 05/14/2010 at 12:04:44
From: Bob
Subject: polynomial long division and negative remainders

Dear Dr. Math,

When elementary students learn long division, the remainders are always
positive. They continue in this way, becoming accustomed to positive
remainders ... until they reach Algebra II, which suddenly confronts them
with negative polynomial remainders.

What is the best way to explain the meaning of negative remainders to
students learning polynomial long division ?

I went through the algorithm (divisor * quotient + remainder = dividend),
showing them the example that ...

   17/5 = 3 + 2/5
   
... is the same as

   4 + (-3)/5.
   
But that didn't really seem to explain what negative remainders mean, so I
felt my explanation was not adequate.

Thanks for your help.

Date: 05/14/2010 at 13:46:50
From: Doctor Peterson
Subject: Re: polynomial long division and negative remainders

Hi, Bob.

When we do long division of numbers, our goal is to have a remainder that
is positive and less than the divisor:

  A = D * Q + R  with 0 <= R < D

where A is the dividend, D is the divisor, Q is the quotient, and R is the
remainder.

When it comes to polynomials, however, we can't say that one polynomial is
less than another; that would depend on the value of x. What we CAN do is
to compare the DEGREE of two polynomials. And when we divide polynomials,
our goal is to have a remainder of DEGREE less than that of the divisor:

  A(x) = D(x) * Q(x) + R(x)  with 0 <= deg(R) < deg(D)

So in this context, whether the remainder (or, for that matter, a term of
the quotient) is positive or negative is irrelevant. The degree of -2x + 1
is less than the degree of 3x^2 + 5x - 4, so we would accept the former as
a remainder if we are dividing by the latter.

It may also be helpful to note that in numbers (written in positional
base-ten notation), each digit represents the coefficient of a power of
10, and must be positive and less than 10. In polynomials, each term
represents a power of x, and the coefficient need only be a number. Sign
doesn't matter. That is the difference between numerals and polynomials,
and it plays out in how we do long division in each context.

If you have any further questions, feel free to write back.

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

Date: 05/14/2010 at 16:47:26
From: Bob
Subject: Thank you (polynomial long division and negative remainders)

Hello Dr. Peterson,

Thank you so much for your prompt reply to my polynomial long division and
remainder question. I appreciate your time and reply!

I have used Dr. Math many, many times before, but never to ask a question.
This was a great process. I may be back!

Thanks again.

Bob Siefken