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

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.

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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.

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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.

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Just adding more to this answer. :) You can find polynomial long division in an Algebraic Structures course, which is similar to Abstract Algebra. Moreover, polynomial long division does occur in algebraic extensions of fields, most importantly, in Galois Field extensions. Ex: Gal($\mathbb{C}/\mathbb{R})$. :) And like the person below me answered, it is useful in Coding Theory, as well...like in Cryptography. :) I'm in this course now, and we just started talking about Galois Theory, so it's fresh in my mind. :) –  user61752 Mar 8 '13 at 3:21

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.

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(+1) Very nice! –  Asaf Karagila Mar 8 '13 at 3:18