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… )


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

The Difference Between a Quartic, and a Quadric

I’ve talked to people who did not distinguish, between a Quartic, and a Quadric.

The following is a Quartic:

y = ax4 + bx3 + cx2 + dx + e

It follows in the sequence from a linear equation, through a quadratic, through a cubic, to arrive at the quartic. What follows it is called a “Quintic”.

The following is a Quadric:

a1 x2 + a2 y2 + a3 z2 +

a4 (xy) + a5 (yz) + a6 (az) +

a7 x + a8 y + a9 z – C = 0

The main reason quadrics are important, is the fact that they represent 3D shapes such as Hyperboloids, Ellipsoids, and Mathematically significant, but mundanely insignificant shapes, that radiate away from 1 axis out of 3, but that are symmetrical along the other 2 axes.

If the first-order terms of a quadric are zero, then the mixed terms merely represent rotations of these shapes, while, if the mixed terms are also zero, then these shapes are aligned with the 3 axes. Thus, if (C) was simply equal to (5), and if the signs of the 3 single, squared terms, by themselves, are:

+x2 +y2 +z2 = C : Ellipsoid .

ellipsoid

+x2 -y2 -z2 = C : Hyperboloid .

hyperboloid

+x2 +y2 – z2 = C : ‘That strange shape’ .

quadr_3


The way in which quadrics can be manipulated with Linear Algebra is of some curiosity, in that we can have a regular column vector (X), which represents a coordinate system, and we can state the transpose of the same vector, (XT), which forms the corresponding row-vector, for the same coordinate system. And in that case, the quadric can also be stated by the matrix product:

XT M X = C

(Updated 1/13/2019, 21h35 : )

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