r/AskMath Weekly Chat - College Level

Polynomials of the form ax^3+bx+c are called depressed cubics. They are typically easier to solve than general cubics but the algebraic solution for this problem is going to be pretty messy since the only solution is an irrational number (approx. x=-3.152) so we can’t use the Rational Root Theorem and long division to factor the polynomial.

If you only need an approximation of the answer something like Newton’s method would converge quickly to the solution.

Getting an exact solution with algebra is more difficult but the following method can be used to solve depressed cubics. Consider the identity

(e + f)^3 -3ef(e + f) - (e^3 + f^3) = 0

Compare this to the given equation x^3 - 2x + 25 = 0. Letting x = e + f and matching coefficients gives us the following three equations

x = e + f , -3ef = -2 , e^3 + f^3 = -25

Solving the second equation for f would give f = 2/3e. Substituting this into the third equation and multiplying both sides by e^3 will give a quadratic polynomial in terms of e^3. Using the quadratic formula we can find e^3, then cube root it to get e. Once you have e use the second equation again to find f. Finally since we had x = e + f you can add these values to find the solution. (It should be approx. x=-3.152)
Hi there! Is anybody here?
x\^3-2x+25=0 I am searching solution for this problem.
full soution method. answer is not the priority
There's probably a better place for this, but I can't find one....

I want to do some simulations of a Swiss-system tournament, for ice hockey. Context: I referee and scorekeep kids' hockey (called minor hockey here in Canada) and adult recreational hockey (called "beer league"). We've just gotten through about 10 weeks' worth of kids' spring tournaments. The teams are subdivided into age groups (e.g. kids born in 2010 and 2009 might be in the same group, or even just one age group for kids born in 2010 and another for 2009). The tournament might have 10 (or more!) teams in one group, and they'll advertise a guarantee that each team will play at least 4 games, so they might make two pools of five teams each, have a round-robin in each pool, and then have the two pool winners play for gold and the two pool runners-up play for bronze. Not always going to give you optimal results.

I was thinking a Swiss system as is often used in chess or go might provide better results, but I want to run some simulations to be able to demonstrate it. Especially, I would need to test initial pair-ups, which vary among different competitions. For example, some rank the entrants by an existing rating (e.g. a FIDE rating), and then make the first round (e.g. for 16 contestants) 1 v 9, 2 v 10 ... 8 v 16.  Others "rank" them randomly and then go 1 v 2, 3 v 4 ....

Are there any good papers/discussions on when to use what? Any good simulation software out there (even if it's just well-laid-out Excel spreadsheets)? Or should I be barking up a different tree?