The probability certain random quadratics have real roots

Early in high school algebra, quadratics chosen as examples by teachers and textbooks alike tend to have integer coefficients and to factorise over the integers. This can give the misleading impression that such quadratics are the norm. As students progress into calculus and begin regularly seeing quadratics that are not as ‘nice’, we hope they become disabused of this notion. Indeed, even if the coefficients of the quadratic are integers, the probability that the quadratic factorises over the integers tends to zero as the range from which the integers are drawn grows (see [1]). But what if we ask about a behaviour less restrictive than factorising, say merely having real roots? This is the problem that concerns this Article.

Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes

Presented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation $$x^4 + a x^3 + b x^2 + c x + d = 0$$ x 4 + a x 3 + b x 2 + c x + d = 0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. The presented classification of the roots of the quartic equation is particularly useful in situations in which the equation stems from a model the coefficients of which are (functions of) the model parameters and solving cubic equations, let alone using the explicit quartic formulæ , is a daunting task. The only benefit in such cases would be to gain insight into the location of the roots and the proposed method provides this. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the “sub-quartic” $$x^4 + ax^3 + bx^2$$ x 4 + a x 3 + b x 2 with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.

Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools

Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p(x,y) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools.

Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

Numerical Solution of Nonlinear Equations in Maple

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

Passion’s Fictions

The final chapter consolidates the implications of the foregoing argument for the interpretation of early modern literature, in part by returning to the start of the story, in Shakespeare; but it approaches Shakespeare by way of an eighteenth-century phenomenon: the rise of works of “character criticism” represented for example by William Richardson’s essays. Eighteenth-century character criticism has long been seen as a new way of reading Shakespeare, even the intrusion of something foreign to Shakespeare’s plays. The word for that foreign element is often “psychology,” especially as allied to reading practices associated with the novel. This chapter argues that the real roots of character criticism lie in much older theories of the passions. The psychology at work is not nearly as new as has been claimed, as can be seen by contrasting Richardson’s essays with one of the books he cites: Edmund Burke’s treatise on the sublime and beautiful. The chapter then circles back to Shakespeare’s Hamlet, arguing that Shakespeare’s plays already contain the elements of a psychology: an externalist psychology grounded in rhetoric and its account of the circumstantial mimesis of actions as an instrument of the knowledge of the passions. Shakespeare’s plays could become the material for a science of the passions because in some sense they already were: instances of a circumstantial knowledge of the passions produced according to principles first theorized by rhetoric, which themselves shaped the new sciences of the mind that developed in the seventeenth and eighteenth centuries.

Real Root Polynomials and Real Root Preserving Transformations

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial p with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real. Let n ≥ 1 and P n be the vector space of all polynomials of degree n or less with real coefficients. In this article, we give explicit forms of polynomials in P n such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on P n which preserve real roots of polynomials in a certain subset of P n .