What branches of math make frequent use of polynomial long division?

Question

I'm reviewing basic algebra right now, as part of a larger math review. I majored in math (undergraduate), and I'm surprised how unfamiliar I am with polynomial long division. I'm sure I've done it at some point in the past, though maybe not since high-school. If it came up at all in my college courses, it wasn't frequent enough for me to remember.

Anyway, I'm left wondering how often it comes up and in what context. When does it show up as a useful tool beyond solving problem sets? I don't doubt that it does, I'm just curious where.

Answer 1

Polynomial long division is used in error-control coding theory and practice. Perhaps the most common use is for error detection via cyclic redundancy check (CRC) methods. Data transmitted as packets on the Internet is followed by a $32$-bit check sum called a CRC checksum. The check sum bits are chosen so that the entire sequence of transmitted bits, which can be thought of as a polynomial with coefficients in the field $\mathbb F_2$, is a multiple of the CRC polynomial of degree $32$. The receiver checks whether the received sequence of bits is in fact a polynomial divisible by the CRC polynomial by carrying out a polynomial long division. If the remainder is zero, the data portion is accepted. If the remainder is nonzero, the data is discarded and a re-transmission of the packet is requested.

For completeness I will say that at the transmitter, the CRC check sum bits are also found by doing a polynomial long division. More details can be found here. If people have ideas as to how these ubiquitous calculations can be speeded up, many engineers will be happy to hear from you.

Answer 2

The theory of polynomial long division is very relevant in the construction of algebraic extensions of fields, though you may not need to actually long divide any polynomials, just understand how it works.

Answer 3

One case where polynomial long division shows up is in the study of Transfer Functions. You will encounter this if you take any Signals and Systems, Signal Processing or Control Theory course.

Additionally, polynomial long division is encountered in computational algebraic geometry, and in particular in the context of Hilbert series computation.

Answer 4

In my case I think it mainly came up when integrating rational functions, which was a core topic in 2nd semester calculus. To a lesser extent, integrating rational functions also came up in courses such as elementary (ordinary) differential equations, engineering separation of variables PDE courses, "advanced mathematics for engineering" courses, etc. I believe I may have divided polynomials in a complex variables course as well.

However, by far the most use I made of dividing polynomials was when helping friends with lower level math (for free), tutoring students in lower level math (for pay), and teaching lower level math classes. Indeed, with the exception of one year in the late 1980s (when I taught arithmetic and geometry in a high school), I think I may have taught long division (or at least strongly reviewed it for students) in at least one class each semester for over 20 years of teaching . . .

Answer 5

So outside of modern algebra and one or two other exotic fields it is no use at all, as partial fractions with degree of numerator >= degree of denominator can be dealt with by other simpler and more general methods. It's time it was kicked out of the school math curriculum. Here is an example:

$(x^3 + 3x^2 + 2x +1)/(x^2 - 4x + 5)$ It is OBVIOUS that an attempt to write this as $(Ax + B)/(x^2-4x+5)$ requires a linear part as well, call it $Cx + D$. So rewrite $Cx + D + (Ax + B)/(x^2-4x+5)$ as $({\rm something})/(x^2-4x+5)$, and 4 linear equations appear, in $A, B, C$ and $D$. Just what is needed to match the original numerator. Try with linear factors $(x - 5)(x - 1)$ in the denominator instead. You still need the $Cx + D$, and it is still fairly obvious. Needless to say, the $Cx + D$ is the quotient of the polynomial division!