Polynomials (1 Viewer)

[DaGr81]

hey peoples help me out on this question
question 16 arnold and arnold exercise 4.1

if P(x) = 1 - x + x^2/2! - .......+ (-1)^n x^n/n! , show that p(x) has no multiple zero for n more than equal to 2.


thanks

 

[spice girl]

Originally posted by DaGr81
hey peoples help me out on this question
question 16 arnold and arnold exercise 4.1

if P(x) = 1 - x + x^2/2! - .......+ (-1)^n x^n/n! , show that p(x) has no multiple zero for n more than equal to 2.


thanks

P'(x) = -1 + x - x^2/2! + x^3/3! - ... + (-1)^n-1 x^(n-1)/(n-1)!
= -P(x) - (-1)^n x^n/n!

now when P'(x) = P(x), - (-1)^n x^n/n! = 0
=> x = 0
but when this happens, P(x) =/= 0

thus the condition P'(x) = P(x) = 0 is never satisfied

thus no multiple roots.

 

[freaking_out]

...n more than equal to 2...

if there are no multiple roots, i am wondering how do u incoparate the quoted condition.

 

[wogboy]

You shouldn't have to worry about that condition n >= 2, since if the theorem is true for all n, as proven, then it is automatically true for n>=2.