Hello!
new user to this cool website
I have a question on characterising the roots of n-degree polynomials. the polynomial comes from an eigenvalue problem, and the goal is to find relations between the roots of the polynomial (without finding the analytical solution).
to this regard, I am stucked with the following problem.
there is a 4-degree monic polynomial like that:
x^4 -a*x^3 + b*x^2 -a*(c^-1)*x + c^(-2),
where {a,b,c} are known parameters (real numbers).
in the article I am reading, the author rewrites the polynomial as such:
(x-x4)*(x-x3)*(x-x2)*(x-x1),
where {x4,x3,x2,x1} are the four roots of the polynomial.
and then he simply says "because of the symmetry of the polynomial, the four roots
satisfy: x1=(c*x2)^-1 and x3=(c*x4)^-1 "
What I understand: the 4-degree polynomial has 4 roots, so (1) can be rewritten as in (2). I understand that plugging the suggested permutation works out in the polynomial.
What I don’t understand: How did the author find the permutation? I cant find a way to come up with them.
I did a simple exercise with a second order monic polynomial and found a permutation for the roots (such as x1 is affine in x2), but I could only do so AFTER I had an analytical solution. I wonder how the guy found his permutations without explicitly solving?
thanks for any help!
truely yours
new user to this cool website
I have a question on characterising the roots of n-degree polynomials. the polynomial comes from an eigenvalue problem, and the goal is to find relations between the roots of the polynomial (without finding the analytical solution).
to this regard, I am stucked with the following problem.
there is a 4-degree monic polynomial like that:
x^4 -a*x^3 + b*x^2 -a*(c^-1)*x + c^(-2),
where {a,b,c} are known parameters (real numbers).
in the article I am reading, the author rewrites the polynomial as such:
(x-x4)*(x-x3)*(x-x2)*(x-x1),
where {x4,x3,x2,x1} are the four roots of the polynomial.
and then he simply says "because of the symmetry of the polynomial, the four roots
satisfy: x1=(c*x2)^-1 and x3=(c*x4)^-1 "
What I understand: the 4-degree polynomial has 4 roots, so (1) can be rewritten as in (2). I understand that plugging the suggested permutation works out in the polynomial.
What I don’t understand: How did the author find the permutation? I cant find a way to come up with them.
I did a simple exercise with a second order monic polynomial and found a permutation for the roots (such as x1 is affine in x2), but I could only do so AFTER I had an analytical solution. I wonder how the guy found his permutations without explicitly solving?
thanks for any help!
truely yours