Hello Pete, First I am sorry to be so sluggish in replying, but I have been preoccupied with medical matters related my own health, which is not so rosy at the moment.
I am not at all expert on your problem, but have given it some thought, so I will just float the ideas that occur to me.
First, there are polynomials with no real solutions, such as x^2 +2, but I will leave them alone for now, and talk about those that do, such as all polynomials of odd degree.
If you find two values of x, say x1,x2, that give values of f(x) with opposite sign, then let x3=(x1+x2)/2, , then calculate f(x3), this must be either zero or have the same sign as either f(x1) or f(x2), so you can make a new pair x3 with x1 or x2 to repeat the cycle, and so on. This converges towards a solution to f(x)=0. If you can't find the suitable x1, x2 to get started I'm not sure what to do, but I hope this helps.
My e-mail is
adwhittle99@gmail.com, so use that if you like, but next week I'm going to be in hospital some of the time. Won't have time for much maths. However, I'll be in touch when I can. I'd like to show a few of my own ideas at some time.
As a matter of interest, where do you live? I am in Barnstaple, in North Devon. I see we're the same age.
