What use are there for learning to divide polynomials?
Two reasons come to mind immediately: First remember, every number is a
polynomial: For example, the number 4321 is: 4xN^3 + 3xN^2 + 2xN^1 + 1xN^0
where N (in our decimal system, N=10) so dividing polynomials is a
generalization of ordinary numerical division. So that in itself is
justification. However, I suspect that isn't what you wanted to know. The
second reason is more "practical". Suppose you were posed with a homework
problem: (6X^3 + 4X^2 - X + 1) is a curve, and you were asked to determine
if that curve could be divided evenly into (2X^2 -1) parts, and how big will
those equal parts be? I don't know the answer "by inspection" but it would
be very easy to find out by dividing the first polynomial by the second and
seeing if there is a remainder. If there is a remainder the answer is NO.
If there is no remainder the answer is YES. If the answer is NO you could
say how many equal parts there would be and how big a "piece" is left over.
I can think of several uses, one of the most important being problem-solving
skills. When dividing one polynomial by another, we seldom have a preset
formula to use. The procedure is fairly simple: factor both numerator and
denominator, and then cancel common factors. This makes it appropriate for
a wide variety of students. Doing a problem, however, requires reasoning
skills. It cannot be done as simple arithmetic.
Although straight factoring can do something similar, polynomial factoring
also prepares a student to interpret graphs. A ratio of polynomials is an
excellent way to introduce zeroes and asymptotes, two very important
concepts in graphical analysis. Being familiar with polynomial ratios
beforehand can greatly help.
In addition, it is a good basis for introduction of limits. It relates the
difference between a function that has a finite limit and a function with an
infinite limit. It is a good example of a function that is missing just the
one point rather than shooting up or down on a graph.
All these things, and more, are often introduced with a ratio of
polynomials. To be familiar with them can be a great asset.
Dr. Ken Mellendorf
Illinois Central College
Dividing polynomials is often done to simplify calculations that are used to
calculate rates (two polynomials that describe things that vary with time
and you are looking to see the relative change between both things --
especially with chemical "rates of reactions").
With the use of computers for modeling, the need to simplify calculations
has gone away to some extent. But the skill is one that checks a true
understanding of the principles of division and a student's ability to take
a skill (such as long division) and expand that skill into a new realm (such
as with polynomials instead of numbers).
I hope this helps! Thanks for your question.
Todd Clark, Office of Science
US Department of Energy
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Update: June 2012