wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> product of the roots
        
(Message started by: tony123 on Nov 7^th, 2007, 11:58am)

Title: product of the roots
Post by tony123 on Nov 7^th, 2007, 11:58am

prove the product of the roots of
50x^50+ 49x^49+ · · · + x + 1 = 0.

is= 1/50.

Title: Re: product of the roots
Post by Aryabhatta on Nov 7^th, 2007, 12:01pm

If I understood the question correctly, this is a basic question and I believe should be in easy.

Title: Re: product of the roots
Post by pex on Nov 7^th, 2007, 12:24pm

on 11/07/07 at 12:01:48, Aryabhatta wrote:

If I understood the question correctly, this is a basic question and I believe should be in easy.

I agree. Just to be thorough:
[hideb]Let the roots be r₁, r₂, ..., r₅₀. By the Fundamental Theorem of Algebra,

50x⁵⁰ + 49x⁴⁹ + ... + x + 1 = 50(x-r₁)(x-r₂)...(x-r₅₀).

Multiplying out, we see that the constant term is 50r₁r₂...r₅₀. As the constant term needs to be equal to one, the result follows.[/hideb]

Title: Re: product of the roots
Post by Grimbal on Nov 8^th, 2007, 1:28am

And what would be the sum of the roots? ::)