Related papers: An alternative quadratic formula
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…
In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic…
This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this…
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
In this paper, we derive the quadratic formula as a consequence of constructively proving the existence of standard and factored forms for general form real quadratic functions. Emphasis is put on connections to graphing of corresponding…
We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding…
We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the…