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The sum and product of an nth degree polynomial 
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#1
Oct1408, 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]_{n1}[/tex] and their product is (1)[tex]^{n}[/tex]a[tex]_{0}[/tex] 2. Relevant equations 3. The attempt at a solution (xc1)(xc2)...(xcn) = x[tex]^{n}[/tex] + (c1+c2+...+cn)x[tex]^{n1}[/tex]....(c1*c2*....*cn) I just need a realistic proof this assumes too much 


#2
Oct1408, 07:15 PM

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



#3
Oct1408, 07:25 PM

P: 45

but how do i know that (xc1)(xc2)...(xcn) = xLaTeX Code: ^{n} + (c1+c2+...+cn)xLaTeX Code: ^{n1} ....(c1*c2*....*cn)?



#4
Oct1408, 10:23 PM

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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.



#5
Oct1508, 05:18 AM

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PF Gold
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#6
Oct1608, 12:27 PM

P: 45

so thats a legit proof then?



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