Abstract
In this chapter, we begin our study of the simplest mathematical expressions, the polynomials. We start with the simplest case: the binomial formula. It is presented here with full arithmetic and historical details, with many identities, and along with its principal, mostly combinatorial, applications including Bernoulli’s derangements. The Division Algorithm for Integers discussed in Section 1.3 leads directly to its polynomial analogue, the Division Algorithm for Polynomials, or polynomial long division, and its offspring, the synthetic division. They reveal a great deal of information about the behavior of polynomials. We accompany these with many examples of (sometimes highly technical) polynomial factorizations. These exhibit beautiful interplays with divisibility properties of integers. Turning to a somewhat more advanced level, we derive the fundamental theorem on symmetric polynomials (leading to a very simple but non-standard derivation of the quadratic formula), the Viète relations, and the Newton–Girard formulas for power sums. Among the many applications of the Viète relations, we give an arithmetic proof of the allegedly most challenging problem ever posted on the International Mathematical Olympiad, in 1988. Finally, we briefly return to the Cauchy–Schwarz inequality, introduced in Section 5.3, in a multivariate setting accompanied by the Chebyshev sum inequality.
“Of course I had progressed far beyond Vulgar Fractions and the Decimal System. We were arrived in an ‘Alice-in-Wonderland’ world, at the portals of which stood ‘A Quadratic Equation.’ This with a strange grimace pointed the way to the Theory of Indices, which again handed on the intruder to the full rigors of the Binomial Theorem.”
in My Early Life by Sir Winston Churchill (1874–1965)
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Notes
- 1.
According to modern terminology, the unknown quantity or quantities within a polynomial (regarded as an expression) are called indeterminates, and they are called variables only when the polynomial is considered as a function. It is, however, widespread to retain the classical terminology and use the word “variable” in both expressions and functions.
- 2.
The modern terminology applied to the much wider class of functions calls a solution of the functional equation f(x) = 0 the zero of the function f.
- 3.
The zero-set ax − by − c = 0 is the generic equation of a line discussed in Section 5.2.
- 4.
The zero-set is the ellipse in normal form to be discussed in Section 8.3.
- 5.
x 2 − d ⋅ y 2 − 1 = 0 is Pell’s equation discussed in Section 2.1.
- 6.
The expansion of this is the Binomial Formula to be discussed in Section 6.3.
- 7.
This polynomial is related to the AM-GM inequality in three indeterminates.
- 8.
The zero-set of this polynomial is the so-called Fermat curverelated to Fermat’s Last Theorem.
- 9.
The first postulate of Euclid’s Elements; see Section 5.1.
- 10.
Also including constant polynomials.
- 11.
A special case (n = 3) was part of a problem in the USA Mathematical Olympiad, 1974.
- 12.
The graphical interpretation of this and similar combinatorial problems is usually termed as “stars and bars,” as advocated by the Croatian-American mathematician Willibald Srećko Feller (1907–1970). In our case, the n one dollar bills are represented by stars, and the separators are the bars.
- 13.
We also reverted to m instead of k for consistency.
- 14.
These identities are referred to by various names. Some reflect the author, some the location of the entries in the Pascal Triangle. For example, iii. is called the Vandermonde-convolution, vi. is the column-sum property, vii. is the SE-diagonal sum property, and viii. is the NW-diagonal sum property. Note, finally, that these identities are interrelated, for example, iii. implies iv., v. implies vi., etc.
- 15.
For a much more detailed account, see the author’s Glimpses of Algebra and Geometry, 2nd ed. Springer, New York, 2002.
- 16.
See also the Crux Mathematicorum (Canadian Mathematical Society), June/July 1978.
- 17.
A variant of this problem is in the Crux Mathematicorum (Canadian Mathematical Society), April 1979.
- 18.
This example is usually treated in multivariate calculus as a simple example of the Lagrange multipliers method. It was also posed as a problem (without the use of calculus) in the MA Θ National Convention, 1987.
- 19.
Sometimes called “Euclidean Division.” Since the proof captures the pivotal step of the associated computational algorithm, usually termed as the “Long Division Algorithm,” and also due to the close analogy with integers, we kept the term “Division Algorithm” for polynomials as well.
- 20.
Expanding and using the Cauchy Product Rule would amount to work out 36 terms.
- 21.
Note the somewhat different layout of the synthetic division in LaTex.
- 22.
See also the Mathematical Olympiad Program, 1997.
- 23.
A similar problem (to calculate only a + b) was in the MA Θ National Convention, 1991.
- 24.
The statement holds for complex roots as well.
- 25.
A similar problem was in the William Lowell Putnam Mathematical Competition, May 1977. An elementary solution (simpler than the one given in the text) is to realize that xy = zw, and make various quadratic expressions in the use of the first equation.
- 26.
A similar problem was in the USA Mathematical Olympiad in 1973.
- 27.
The typical proof uses the completing the square technique.
- 28.
This is the so-called completing the square technique; equivalent to the Quadratic Formula.
- 29.
A special case (n = 63) was a problem in the American Mathematics Competitions, 2002.
- 30.
Although a and b play symmetric roles, the choice of the indeterminate y (and not x) is justified by the geometric content of the problem to be discussed in Section 8.4.
- 31.
A similar problem was in the USA Mathematical Olympiad, 1987.
- 32.
This was a problem in the USA Mathematical Olympiad, 1973; a straightforward solution uses the Newton–Girard formulas.
- 33.
For analogy, the discriminant D of the quadratic polynomial x 2 + px + q is D = (r 1 − r 2)2 = p 2 − 4q, where r 1, r 2 are the roots.
- 34.
This is a generalization of a problem in the Iranian Mathematics Competition, 1997.
- 35.
The special case n = 5 was a problem in the USA Mathematical Olympiad, 1983.
- 36.
In the second equality we can also use the Newton–Girard formula \(p_2(r_1,\ldots ,r_n)=a_{n-1}^2-2a_{n-2}\) along with the Viète relations.
- 37.
Due to its usefulness in some mathematical contest problems, this is sometimes called the Titu–Engel–Sedrakyan inequality after Titu Andreescu (1965 –), Arthur Engel (1928 –), and Nairi Sedrakyan (1961 –).
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Toth, G. (2021). Polynomial Expressions. In: Elements of Mathematics. Undergraduate Texts in Mathematics(). Springer, Cham. https://doi.org/10.1007/978-3-030-75051-0_6
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