Polynomials
Polynomials are mathematical functions that consist of variables raised to non-negative integer powers, combined through addition and multiplication. They have significant applications in various fields, including mathematics, science, and engineering, where they are used to model and analyze data. For instance, polynomial error-correcting codes are essential in technologies that transmit electronic signals, such as space probes and DVD players. Additionally, polynomials are foundational in cryptography, aiding secure data transmission.
In education, polynomials are typically introduced in middle and high school curriculums, with early concepts of functions starting in elementary grades. The study of polynomials has historical roots, with notable contributions from mathematicians across cultures, including those from ancient Babylon to the Middle Ages. Noteworthy theorems, such as the Fundamental Theorem of Algebra, provide insights into the properties of polynomials, including their zeros.
Moreover, polynomials play an important role in modern applications like computer graphics through Bezier curves, demonstrating how simple mathematical ideas can lead to complex and impactful technologies. Overall, polynomials are vital components of mathematical theory and practical applications, reflecting their versatility and fundamental nature in a wide array of disciplines.
Polynomials
Summary: Polynomial functions have long been studied by mathematicians and have interesting and important applications.
Polynomials have a broad array of theoretical and real-world applications and are widely used by mathematicians, scientists, and engineers to mathematically model data and explore many mathematical and scientific concepts. Technologies that transmit electronic signals, ranging from deep space probes communicating with Earth, to home DVD players, commonly use polynomial error-correcting codes, like the Reed–Solomon codes, named for mathematicians Irving Reed and Gustave Solomon. Cryptographic algorithms that help ensure secure data transmission also rely on polynomials to represent and manipulate data. Calculators may use approximations called “Taylor polynomials,” named for mathematician Brook Taylor, for functions like square roots. Civil engineers model and estimate properties, such as volume for lakes and other irregular natural features, with polynomials. Orthogonal polynomials provide the foundation for many multivariate statistical procedures. In twenty-first-century classrooms, polynomials are typically part of advanced middle school or high school curriculums, though linear functions and comparisons of linearity versus nonlinearity are common in middle school, and some of the basic concepts of functions are introduced in the elementary grades.
Early in their mathematical studies, students learn that the graph of the squaring function is a parabola, and that the plot of y=p(x)=x2 is shown in Figure 1, which is the first natural function to consider beyond ones that generate straight lines.
There is an entire family of functions like the squaring function, the cubing function, the fourth power function, and more. If indexed, one could call
the squaring function p2(x) = x2,
the cubing function p3(x) = x3,
the fourth power function p4(xP) = x4,
and, in general, the nth power function pn(x) = xn.
The family of power functions also includes the zero power function p0(x)=1 and the first p1(x)=x1.These power functions are the building blocks of “polynomial functions,” functions that are made from taking sums and constant multiples of power functions. As such, these functions are especially simple because their formulas only involve addition and multiplication. The first understanding of these power functions is generally credited to Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji, who lived c. 1000 c.e. in what is now Iraq. In particular, he made advances in the use of variables and humankind’s ability to think of arithmetic operations on “placeholders,” instead of simply on individual numbers.
Finding the Zeros
Consider this example, p(x)=x3-2x2-4x+8: this function is obtained by taking the cubing function, subtracting twice the squaring function, subtracting 4 times the first power function, and finally adding 8. Regardless of the power functions chosen and the constants multiply by, a polynomial is built. That is, polynomials are functions that have the form
p(x) = anxn + … + a2x2 + s1x + a0
where a0, a1,…, an are real numbers. Provided that an is not zero, it is stated that p is a degree n polynomial; the degree represents the highest power of x that is present. Much of the modern notational perspective on these functions is due to the work of René Descartes, who in the early 1600s did important work that popularized not only the notation above using subscripts and superscripts but also offered a visual perspective on polynomial functions through their graphs.
Going back to the first polynomial example, p(x)=x3-2x2-4x+8, one can rewrite this sum of multiples of power functions in the formula as a product of even simpler functions. Specifically, it is possible to show that
p(x) = x3 − 2x2 − 4x + 8 = (x + 2)(x − 2)(x − 2).
One can easily observe that p(-2) and p(2)=0. Mathematicians call -2 and 2 the zeros or roots ofp(x); since the (x-2) factor, which leads to the zero 2, appears twice, mathematicians say that “2 is a double root” or “2 is a zero of multiplicity two.” The graph of the polynomial in Figure 2 is also enlightening as it shows that the zeros of the function lie where the function crosses or touches the horizontal axis:
If one shifts the graph of the degree 3 polynomial p(x) (in black) slightly up, the new graph (top line in light gray) will have just one real zero, while if one shifts the graph slightly down, the new function (bottom line in medium gray) will have three distinct real zeros. This illustration demonstrates an important fact about degree 3 polynomials: every degree 3 polynomial has 1, 2, or 3 distinct real zeros. Indeed, the Fundamental Theorem of Algebra, which was proved in its earliest form in 1799 by the great mathematician Carl Friedrich Gauss, states that every polynomial of degree n has at most n distinct real zeros.
If one is willing to permit zeros to be complex numbers and count zeros by their multiplicity, a much stronger version of the Fundamental Theorem of Algebra (which was also known to Gauss) can be proved: every polynomial of degree n has exactly n zeros, provided one counts them according to their multiplicity and allows zeros to be complex. The Fundamental Theorem of Algebra asserts only that n roots of a polynomial function of degree n exist; it does not tell what those roots are.
Quadratic, Cubic, and Quartic Formulas
The search for the zeros of polynomial functions attracted many great minds. The quadratic formula, which calculates the zeros of any degree 2 polynomial, was understood in certain forms by Babylonian mathematicians as early as 2000 b.c.e. The quadratic formula asserts that in order for ax2+bx+c=0, it must be the case that
For cubic equations and their roots—finding where a polynomial of degree 3 is zero—it took another 3500 years for mathematicians to fully understand the situation. Following contributions from ancient Greeks, Indians, and Babylonians, as well as Persians in the eleventh and twelfth centuries, a group of Italian mathematicians in the 1500s (Scipione del Ferro, Niccolo Tartaglia, and Gerolamo Cardano) proved that there is a cubic formula. In other words, based on the coefficients of a degree 3 polynomial, there is a very complicated formula involving cube roots that calculates the locations of the polynomial’s zeros.
Mathematicians were able to take these discoveries a step further. Near the mid-1500s, Ludovico Ferrari found a way to solve quartic equations. This quartic formula is incredibly complicated and represents a major feat in the understanding of polynomial functions. Interestingly, these general formulas cease to exist beyond polynomials of degree 4. In 1824, Neils Abel and Paolo Ruffini published a theorem, based on the work of Evariste Galois, proving that there was no general formula for the roots of a degree 5 polynomial or higher. This latter work on polynomials ended up founding an entire new branch of mathematics called modern algebra. Sometimes in mathematics, the quest to solve one problem leads to a whole host of other interesting problems or even a new collection of coherent ideas.
Applications
Polynomial functions demonstrate all sorts of interesting patterns and properties and have long been studied because they are interesting in their own right. But even more than this, polynomials play important roles in other areas of mathematics and in applications. For example, polynomial functions spawned the subject of modern algebra, and key ideas in modern algebra are used in the field of public key cryptography—the science of keeping important information private in such essential settings as Internet commerce.
A more direct application of polynomial functions comes in the design of fonts that appear on computer screens. So-called Bezier curves, named for mathematician Pierre Bezier, are degree 3 polynomial functions that can be easily spliced together to form elegant shapes. For instance, at right is the letter S in the Palatino font.
Each piece of the S—the portion of the curve between consecutive squares that represent points on the curve—consists of a degree 3 parametric polynomial. There is deep and elegant mathematics behind why Bezier curves work so well and why they are particularly suited to computer graphics. This is just one example of how substantial ideas and applications in mathematics often emerge from simple beginnings.
Polynomial functions are the simplest of all functions, can be used to approximate more complicated functions that are not polynomials, and often emerge in important applications. They are indeed some of the key building blocks of mathematics.
Bibliography
Barbeau, Edward. Polynomials. New York: Springer, 1989.
Kalman, Dan. Polynomia and Related Realms. Washington, DC: Mathematical Association of America, 2009.
Kushilevitz, Eyal. Some Applications of Polynomials for the Design of Cryptographic Protocols. Berlin: Springer-Verlag, 2002.
Strogatz, Steven. “Power Tools.” New York Times. http://opinionator.blogs.nytimes.com/2010/03/28/power-tools/.