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Polynomials

Graph of a parabola and a cubic equation on a chalkboardThe word polynomial means many terms (literally, many names).  In modern usage, however, this expression includes monomials (one term, such as 5 or 3x or -7x³), binomials (two terms, such as x² – 3 or 2x³ + 9x), and trinomials (three terms, such as x² + 5x – 3).  Each term can have more than one variable, but each variable may only have a power that is a natural number (0, 1, 2, 3, …).  No other functions are allowed.  So, terms such as -4x³, 7xy², and 9zy³z²/7 are all valid terms (think of them as the ‘nomials’ of which you may have ‘many’).

Because most of the material on this page is designed for high school (and some early university) use, I will be restricting my usage, mostly, to using one variable, so I will be writing of y being a function of x [y = f(x)] or of y being a polynomial in x [y = P(x)].  The examples to the left, where you see the graph of a quadratic and a cubic equation, will be typical.  When expressed in this way, such polynomials are also functions. In other words, for any given x value, there is only one y value.  These polynomials also have some very nice properties, as we shall see.  All these things make them easier to study than other curves, but they still have some nice surprises in store for us!  Please note that I am aware that there is a difference in meaning between indeterminates and variables but, for ease of use, I will use the term variable because that is what most high school students will be familiar with.

It is difficult to pin down when this term was first used. Mathematicians had been studying them for many years, but without using this particular term to describe the entire collection.  There are claims that the term polynomial was first used by the brilliant Franch mathematician, François Viète (1540-1603), but I have been unable to validate them (please see the first drop-down item below).

[As an aside, the way that we write our algebra has improved immensely since Viète’s day.  Robert Recorde had created the = sign in 1557, but it was yet to become popular. Karr, Massey and Gustafson, in their Intermediate Algebra share that, where we would write (x + 1)³ = x³ + 3x² + 3x + 1, Viète would have written  cubus aequalis x cubus + x quad. 3 + x in 3 + 1.  And you think modern notation is difficult!]

As a more formal definition, a polynomial in one variable (or, more correctly, indeterminate) can be written in the form a subscript n times x to the power n plus a subscript n minus one times x to the power n minus one and so on where a subscript zero, up to a subscript n are numbers, being elements of a ring, so that the normal four operations (addition, subtraction, multiplication and division) apply to the variable/indeterminate, the coefficients, and the terms.  The graphs of polynomials of this kind form a particular family of curves which share some interesting characteristics.

At various times, as you study this material, you may with to refer to a useful site I found which provides a lot of information about the History of Polynomial Equations.

Analysing and Creating Polynomials Using the Babylonian Differencing Technique

How to Graph Polynomials and How to Construct Their Equations From Their Graphs

Notes Concerning the Origin of the Term 'Polynomial'

A number of online sources claim that François Viéta (1540-1603) first used the term polynomial, and quote Cajori 1919, page 139 as a reference:

They also observe that the word is (first) found in English in Samuel Jeake's (1623-1690) Arithmetic (1674): "Those knit together by both Signs are called ... by some Multinomials, or Polynomials, that is, many named." They all provide a reference of OED2 (second edition of the Oxford English Dictionary).

As I mentioned above, althought Viète certainly worked with polynomials, I cannot trace any reference that he called them by that name.  I managed to find an online copy of Florian Cajori's (1859-1930) book, A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse (1919), but could not find any such reference on page 139 or anywhere else in it.  So, I remain puzzled.

Page 273 from Samuel Jeake's book showing the first use of the word polynomial in EnglishThe reference for the first use of the word polynomial in English (identified by the Oxford English Dictionary) was much easier to trace.  Although I could not find a reference to a 1674 publication, Wikipedia states that Jeake's principal work was Arithmetick Surveighed and Reviewed published in four books in 1696. You may view the relevant page 273 (see right) via Google Books (and may download a PDF copy of the entire book by clicking on the [EBOOK - FREE] button).  I also located a facsimile of his 1701 book, A Compleat Body of Arithmetick in Four Books (again via Google Books), which is actually the same publication. A facsimile of this 1701 publication is available for purchase if you so desire.

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