Factoring Quartic Polynomials: A Lost Art

@article{Brookfield2007FactoringQP,
  title={Factoring Quartic Polynomials: A Lost Art},
  author={Gary Brookfield},
  journal={Mathematics Magazine},
  year={2007},
  volume={80},
  pages={67 - 70}
}
  • G. Brookfield
  • Published 1 February 2007
  • Mathematics
  • Mathematics Magazine
(2007). Factoring Quartic Polynomials: A Lost Art. Mathematics Magazine: Vol. 80, No. 1, pp. 67-70. 

Remainder and quotient without polynomial long division

We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the

In nite families of reciprocal monogenic polynomials and their Galois groups

We prove a new irreducibility theorem for a particular class of polynomials, and we use it to construct in nite families of reciprocal monogenic polynomials. These results extend previous work on

An Interesting Construction Problem

A collection of simple construction problems is presented and conditions for which the construction can be carried out with compass and straightedge are determined.

Galois Groups of Certain Even Octic Polynomials

. Let f ( x ) = x 8 + ax 4 + b ∈ Q [ x ] be an irreducible polynomial where b is a square. We give a method that completely describes the factorization patterns of a linear resolvent of f ( x ) using

Quadratic Residues and the Frobenius Coin Problem

1. G. D. Birkhoff, A set of postulates for plane geometry based on scale and protractor, Ann. Math., 33 (1932) 329-345. 2. B. Bold, Famous Problems of Geometry and How To Solve Them, Dover, 1969. 3.

Galois groups of chromatic polynomials

The chromatic polynomial P ( G, λ ) gives the number of ways a graph G can be properly coloured in at most λ colours. This polynomial has been extensively studied in both combinatorics and

J an 2 02 2 The solutions to single-variable polynomials , implemented and verified in Lean An experience report on learning how to use a computer proof assistant

This work describes the learning experience of a starting Lean user, including a detailed comparison between the authors' work in Lean and very closely related work in Coq, and describes the solutions of quadratic, cubic, and quartic polynomials over certain fields.

Two Families of Monogenic $S_4$ Quartic Number Fields

Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and

Regularization of the boundary integrals in the bem analysis of 3D potential problems

In the conventional boundary element analysis, near-singularities are present in the associated boundary integral equation for problems involving ultra-thin media. For this case, any conventional

The true maximum-likelihood estimators for the generalized Gaussian distribution with p = 3, 4, 5

ABSTRACT The generalized Gaussian distribution with location parameter μ, scale parameter σ, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p =

References

SHOWING 1-7 OF 7 REFERENCES

On the Solution of the Real Quartic

(1966). On the Solution of the Real Quartic. Mathematics Magazine: Vol. 39, No. 1, pp. 28-30.

Can This Polynomial Be Factored? Two-Year

  • College Math. J
  • 1977

Algebra: An elementary text-book.--

Algebra, An Elementary Textbook, Part I

  • 1964

Can This Polynomial Be Factored? Two-Year College Math

  • J
  • 1977