The use of EdPuzzle to learn polynomial factorization in Secondary Education

INTRODUCTION. One of the main objectives of Mathematics education is to motivate students since their interest in Mathematics is very low in many cases and, in others, even null. The use of different technologies has grown a lot in the last decades and several authors in the area have demonstrated their effectiveness in the classroom. In addition, the use of different Information and Communication Technologies, which students can use in their own homes, is also growing. METHOD. In this study, the use of the EdPuzzle application is presented, under the paradigm of the Flip Learning methodology in the third year of the Secondary Education Level for learning polynomial factorization in a school in Spain in two different courses, taught by the same teacher. To carry out the experience, the average grades of two exams have been taken, the first one related to polynomials and the basic concepts, used as a pretest since it has the main basic concepts that the student must know before continuing, and another related to the polynomial factorization, used as a posttest, and the scores obtained by the group that used EdPuzzle were comparedwith those that did not use it. RESULTS. Comparison of both groups shows that the scores are significantly higher in the group that used EdPuzzle in the posttest. DISCUSSION. The Cohen’s d effect size obtained was almost medium, and the questionnaire answers were positive, aspects that make EdPuzzle a tool to be considered in the teaching-learning process of Mathematics.

On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L=Kα, we give a method to describe all absolute values of L extending ∣∣, where K is a discrete rank one valued field.

Computing one billion roots using the tangent Graeffe method

Let p be a prime of the form p = σ2 k + 1 with σ small and let F p denote the finite field with p elements. Let P ( z ) be a polynomial of degree d in F p [ z ] with d distinct roots in F p . For p =5 · 2 55 + 1 we can compute the roots of such polynomials of degree 10 9 . We believe we are the first to factor such polynomials of size one billion. We used a multi-core computer with two 10 core Intel Xeon E5 2680 v2 CPUs and 128 gigabytes of RAM. The factorization takes just under 4,000 seconds on 10 cores and uses 121 gigabytes of RAM. We used the tangent Graeffe root finding algorithm from [27, 19] which is a factor of O (log d ) faster than the Cantor-Zassenhaus algorithm. We implemented the tangent Graeffe algorithm in C using our own library of 64 bit integer FFT based in-place polynomial algorithms then parallelized the FFT and main steps using Cilk C. In this article we discuss the steps of the tangent Graeffe algorithm, the sub-algorithms that we used, how we parallelized them, and how we organized the memory so we could factor a polynomial of degree 10 9 . We give both a theoretical and practical comparison of the tangent Graeffe algorithm with the Cantor-Zassenhaus algorithm for root finding. We improve the complexity of the tangent Graeffe algorithm by a factor of 2. We present a new in-place product tree multiplication algorithm that is fully parallelizable. We present some timings comparing our software with Magma's polynomial factorization command. Polynomial root finding over smooth finite fields is a key ingredient for algorithms for sparse polynomial interpolation that are based on geometric sequences. This application was also one of our main motivations for the present work.