Abstract
Factoring quadratics over Z is a staple of introductory algebra and textbooks tend to create the impression that doable factorizations are fairly common. To the contrary, if coefficients of a general quadratic are selected randomly without restriction, the probability that a factorization exists is zero. We achieve a specific quantification of the probability of factoring quadratics by taking a new approach that considers the absolute size of coefficients to be a parameter n. This restriction allows us to make relative likelihood estimates based on finite sample spaces. Our probability estimates are then conditioned on the size parameter n and the behavior of the conditional estimates may be studied as the parameter is varied. Specifically, we enumerate how many formal factored expressions could possibly correspond to a quadratic for a given size parameter. The conditional probability of factorization as a function of n is just the ratio of this enumeration to the total number of possible quadratics consistent with n. This approach is patterned after the well-known case where factorizations are carried out over a finite field. We review the finite field method as background for our method of dealing with Z [x]. The monic case is developed independently of the general case because it is simpler and the resulting probability estimating formula is more accurate. We conclude with a comparison of our theoretical probability estimates with exact data generated by a computer search for factorable quadratics corresponding to various parameter values.
Highlights
Factoring quadratics over is a staple of introductory algebra and textbooks tend to create the impression that doable factorizations are fairly common
We enumerate how many formal factored expressions could possibly correspond to a quadratic for a given size parameter
We wish to have a more granular understanding of the way in which factorability depends on commensurability of coefficients. Our approach to this question is motivated by solving the factorization probability problem in the context of finite fields, which we review below
Summary
Dealing with the infinite number of choices available for coefficients is a problem We sidestep this obstacle by adapting the method used to determine the probability of factorization of quadratics over finite fields. We establish a cutoff, or size parameter n, for the absolute value of any coefficients appearing in any of the quadratics we wish to study This makes the number of quadratics under consideration finite as well as the number of formal factored expressions that could possibly yield such a quadratic. The classical probability is just the ratio of the number of admissible factored expressions to the total number of quadratics which conform to the cutoff This probability P (n) is, a conditional probability given that the coefficients do not exceed n in absolute value. Factoring Over GF pn Suppose we are given a random monic quadratic over the finite field GF ( p)
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