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
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