# The sum and product of an nth degree polynomial

1. ### phyguy321

45
1. The problem statement, all variables and given/known data
Suppose f(x) $$\in$$ Complex[x] is a monic polynomial of degree n with roots c1,c2,...cn. Prove that the sum of the roots is -a$$_{n-1}$$ and their product is (-1)$$^{n}$$a$$_{0}$$

2. Relevant equations

3. The attempt at a solution
(x-c1)(x-c2)...(x-cn) = x$$^{n}$$ + (c1+c2+...+cn)x$$^{n-1}$$....(c1*c2*....*cn)

I just need a realistic proof this assumes too much

2. ### Dick

25,887
In what way do you think that's assuming too much? Do you know the Fundamental Theorem of Algebra?

3. ### phyguy321

45
but how do i know that (x-c1)(x-c2)...(x-cn) = xLaTeX Code: ^{n} + (c1+c2+...+cn)xLaTeX Code: ^{n-1} ....(c1*c2*....*cn)?

4. ### Dick

25,887
Count powers of x. There's only one way to make x^n and x^0. There are n ways to make x^1. You just imagine multiplying it out.

5. ### HallsofIvy

40,675
Staff Emeritus
Because you know how to multiply polynomials?

6. ### phyguy321

45
so thats a legit proof then?