In certain situations, Maxima can actually solve a sextic equation.

For readers who don’t know, a sextic equation is a polynomial of the 6th degree. As the subject line suggests, recent versions of Maxima can find symbolic solutions to those, if used correctly, and, if the sextic actually has ‘an exact, analytical solution’, which is also referred to sometimes as ‘a symbolic solution’.

Whether these analytical solutions are actually more useful than numeric approximations, remains an unanswered question.

What has happened to me is, that I’ve tried to use the method shown below, to cause Maxima to display the solution, and that due to what amounted to a typo, I had given it a polynomial which was visually similar to the one shown, but which was also different in some small way, so that the only solution which Maxima displayed, was the original polynomial, thus implying that Maxima was not able to solve an altered one. The reason this happened is easy to explain…

Not all polynomials of the 6th degree actually have an analytical solution. If given an example that does not, Maxima will fail to display one. All polynomials of the 4th degree actually have an analytical solution, but it may easily be too complex for consumer-grade Computer Algebra Systems (CAS) to output. But, by the time the user is asking a CAS to solve a cubic, he should be able to expect this form of a solution to be output.

The sextic below is actually the product of two cubics, which also explains why Maxima was able to solve it. The reader will need to enable JavaScript:

  • From my site, And
  • From MathJax.org,

To be able to view the worksheet:

 


 

(Updated 7/04/2020, 13h30… )


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How an exact solution can sometimes be found, without using the general solution.

One of the facts which I’ve been writing about is, that the general solution to a polynomial of a degree higher than (4), that is expected to produce Algebraically exact results, cannot be used because none exists. At the same time, I make a distinction between an exact solution, and the general solution. This distinction can also be explained in greater detail…

We are sometimes given a polynomial, which has at least one “rational root”, meaning a root that can be stated either as a whole number, which is actually referred to as an “integer”, or as a fraction. The following is an example:

x^3 -3*x^2 -2*x + 6 = 0

In this case it can be observed, that the coefficient of (x^3), which is not stated, corresponds to a (1), and that the constant term, which is visible as (+6), is an integer. What can be done here, is that all the factors of (6) can be used positively and negatively – not only the prime factors – and plugged in to see whether they do in fact constitute one root. Again, they do if and only if the equation is satisfied as resulting in zero.

Thus, as potential candidates, ±1, ±2, ±3, ±6 can all be tried.

(Updated 3/2/2019, 16h30 … )

Continue reading How an exact solution can sometimes be found, without using the general solution.

How the general solution of polynomials, of any degree greater than 2, is extremely difficult to compute.

There are certain misconceptions that exist about Math, and one of them could be, that if a random system of equations is written down, the Algebraic solution of those equations is at hand. In fact, equations can arise easily, which are known to have numerical answers, but for which the exact, Algebraic (= analytical) answer, is nowhere in sight. And one example where this happens, is with polynomials ‘of a high degree’ . We are taught what the general solution to the quadratic is in High-School. But what I learned was, that the general solution to a cubic is extremely difficult to comprehend, while that of a 4th-degree polynomial – a quartic –  is too large to be printed out. These last two have in fact been computed, but not on consumer-grade hardware or software.

In fact, I was taught that for degrees of polynomials greater than 4, there is no general solution.

This seems to fly in the face of two known facts:

  1. Some of those equations have the full number of real roots,
  2. Computers have made finding numerical solutions relatively straightforward.

But the second fact is really only a testament, to how computers can perform numerical approximations, as their main contribution to Math and Engineering. In the case of polynomials, the approach used is to find at least one root – closely enough, and then, to use long division or synthetic division, to divide by (1 minus that root), to arrive at a new polynomial, which has been reduced in degree by 1. This is because (1 minus a root) must be a factor of the original polynomial.

Once the polynomial has arrived at a quadratic, computers will eagerly apply its general solution to what remains, thereby perhaps also generating 2 complex roots.

In the case of a cubic, a trick which people use, is first to normalize the cubic, so that the coefficient of its first term is 1. Then, the 4th, constant term is examined. Any one of its factors, or the factors of the terms it is a product of, positively or negatively, could be a root of the cubic. And if one of them works, the equation has been cracked.

In other words, if this constant term is the product of square roots of integers, the corresponding products of the square roots, of the factors of these integers, could lead to roots of the cubic.

(Updated 1/12/2019, 21h05 … )

Continue reading How the general solution of polynomials, of any degree greater than 2, is extremely difficult to compute.