Finding a Unique Solution for a Polynomial Equation

[robbycon]

So I have an equation:
h(x) = \sum a_{i}x^{i} from i=0 to d.

I know a_{i} and x.
I am trying to prove that there is a y where g(x) = \sum a_{i}y^{i} from i=0 to d, g(x) = h(x), and y does not = x.

How do I do this? Sorry for the bad use of Latex.

 

[Office_Shredder]

In general? You can't, for example, the polynomial x³ is 1-1. It's unclear what you mean here, g(x) is not actually a function of x, and in fact seems to be h(y). So what you really want is to find x and y so that h(x)=h(y) right?

 

[robbycon]

Yes, sorry. That was a typo. That is g(y), not g(x).

 

[HallsofIvy]

It's still not clear what you want to prove. If you have h(x)= \sum_{i=0}^d a_i x^i and you replace the variable x by any y, you get h(y)= \sum_{i=0}^d a_iy^i. It is the same function, just written differently.

If you mean x and y to be specific numbers and want to prove that there exist y\ne x such that h(y)= h(x), you can't- it is not, in general, true. As Office Shredder says, polynomials can be one-to-one. His example of h(x)= x^{3/sup] shows that.}

 

[robbycon]

I am looking for a y in terms of a_{i} and x^{i} in a solution that contains no polynomial equation. The most important thing I think is to figure out the properties of the summation of a_{i} (which I don't know at all). I know I can't just divide that out, but there has to be a way to separate it from the summation of y.