Register to reply

The sum and product of an nth degree polynomial

by phyguy321
Tags: degree, polynomial, product
Share this thread:
phyguy321
#1
Oct14-08, 06:58 PM
P: 45
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
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
Dick
#2
Oct14-08, 07:15 PM
Sci Advisor
HW Helper
Thanks
P: 25,228
In what way do you think that's assuming too much? Do you know the Fundamental Theorem of Algebra?
phyguy321
#3
Oct14-08, 07:25 PM
P: 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)?

Dick
#4
Oct14-08, 10:23 PM
Sci Advisor
HW Helper
Thanks
P: 25,228
The sum and product of an nth degree polynomial

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
#5
Oct15-08, 05:18 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,533
Quote Quote by phyguy321 View Post
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
#6
Oct16-08, 12:27 PM
P: 45
so thats a legit proof then?


Register to reply

Related Discussions
Polynomial of degree 7 Linear & Abstract Algebra 2
3rd degree polynomial Precalculus Mathematics Homework 1
5th Degree Polynomial Matrix Calculus & Beyond Homework 1
Polynomial of degree 4 Calculus & Beyond Homework 0
Roots of a 4th degree polynomial Precalculus Mathematics Homework 5