The sum and product of an nth degree polynomial

[phyguy321]

Homework Statement

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$$_{0}$$

Homework Equations

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

 

[Dick]

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

 

[phyguy321]

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)?

 

[Dick]

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.

 

[HallsofIvy]

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)?

Because you know how to multiply polynomials?

 

[phyguy321]

so that's a legit proof then?