How to show that a root of the

Jefferson Pacheco

Jefferson Pacheco

Answered question

2022-04-06

How to show that a root of the equation
x(x+1)(x+2).(x+2009)=c

Answer & Explanation

Kaphefsceasiaf8w9

Kaphefsceasiaf8w9

Beginner2022-04-07Added 9 answers

Let denote by
P(x)=x(x+1)(x+2).(x+2009)c
We have
P(0)=P(1)=P(2009)=c
and by Rolle's theorem there's
0>t01>>t20082009
s.t.
P(t0)=P(t1)==P(t2008)=0
Since the polynomial P has the degree 2010 then the degree of P' is 2009 and then the ti are the roots of P'
If c=0 then the roots of P are different from the roots of P' and there is not a multiple root.
If c0, we apply the Rolle's theoem on P' and we find the roots si of P s.t.
t0>s0>t1>>s2007>t2008
and if we have P(α)=P(α)=0 then α=ti0 and then αsii=0,,2007 and hence P(α)0

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