The sum and product of an nth degree polynomial

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

    2. Relevant equations



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

    I just need a realistic proof this assumes too much
     
  2. jcsd
  3. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    In what way do you think that's assuming too much? Do you know the Fundamental Theorem of Algebra?
     
  4. 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)?
     
  5. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    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.
     
  6. HallsofIvy

    HallsofIvy 40,310
    Staff Emeritus
    Science Advisor

    Because you know how to multiply polynomials?
     
  7. so thats a legit proof then?
     
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