Hi,

do you know

1) how to determine roots of polynomial a*x^4+b*x^3+c*x^2+d*x+e=0,

(some method for it), inclusive complex roots

2) this polynomial is 3D, does it make any difference in number of roots?

3) how to display this roots in 3D graph, inclusive complex roots

I'd like to compute and draw it in Maple. Could you write me convenient commands to this problem?

thanks

do you know

1) how to determine roots of polynomial a*x^4+b*x^3+c*x^2+d*x+e=0,

(some method for it), inclusive complex roots

2) this polynomial is 3D, does it make any difference in number of roots?

3) how to display this roots in 3D graph, inclusive complex roots

I'd like to compute and draw it in Maple. Could you write me convenient commands to this problem?

thanks

2)not sure what you mean by this

3) if, say a:=3+8I then to extract real/imaginary parts use:

>Re(a);

>Im(a);

be careful to use "Re" and "Im" exactly i.e. "re" "im" wont work.

First, you have a 4th grade polinomy, with X being the variable, but X is NOT a real variable, it is an imaginary variable (as long as they ask your for the roots, they are supposing the roots exists, so the variable X have to be imaginary), so you have a function value in any (x,I*y) imaginary number, asociated with (x,y) point of the X-Y plane.

Defining z value as the value of the polinomy at any point (x,y), assuming it means polinomial value on (x+Iy), could give you almost what you are searching, a 3D grid, BUT z will be also an imaginary value, in the form zr+I*zi, or (zr, zi) if you prefer, so you have a 4th dimensional graph compossed by all the point in the form (x,y)(zr,zi) having P(x,y)=(zr,zi):

z = (zr+I*zi) = P(x,y) = P(x+y*I) = a*(x+y*I)^4+b*(x+y*I)^3+c*

You need to take out one of those dimensions but without changing the roots, so I propose to define z value as the length of the vector (zr, zi), or the module of the imaginary number zr+I*zi (is the same), so z=sqrt(zr^2+zy^2)=|P(x,y)|

z = |P(x,y)| = |P(x+y*I)| = |a*(x+y*I)^4+b*(x+y*I)^3+c

Now you have a 3D grid that represent the polinomy, and 4 points (x,y) that are the roots of the polinomy: They are just the points in the 3D grid where the grid lays on the X-Y plane, having z=0, so plotting them is just a matter of highlighting 4 points in the 3D grid.

There are also other ways to represent P(x,y) without using z=|P(x,y)|, but as they involve 4 dimensions, visualizing can be "obscure".

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