I am in the middle of writing an essay for my Using Radicals to Solve module, and I'm sorry if this is a silly question, but I cannot find any answers for this anywhere so any help would be appreciated!
All I have so far is the definition of a depressed polynomial and the advantages of the depressing process.
I can't seem to figure out what effect the translation $y=x-e$ will have on higher degree polynomials, and why that would be helpful when solving them
Any advice would be so appreciated!
A general answer could be : having one term less, one has one parameter less to cope with.
Let us give an illustration. Let us consider the general 3rd degree polynomial :
$$x^3+ax^2+bx+c$$
vs. its depressed form, traditionaly written :
$$x^3+px+q$$
In the first case, the discriminant is
$$a^2b^2+18abc-4b^3-4a^3c-27c^2$$
whereas in the second case it is plainly :
$$-(4p^3+27q^2).$$
(Let us recall that the sign of the discriminant of a third degree polynomial indicates if it has 1 or 3 real roots).