Consider the following polynomial P ( z ) = ( z &#x2212;<!-- −

codosse2e

codosse2e

Answered question

2022-05-31

Consider the following polynomial
P ( z ) = ( z a ) ( 1 a z ) b z + Q ( z ) , with a ( 0 , 1 ) and b R + ,

Answer & Explanation

aniizl

aniizl

Beginner2022-06-01Added 12 answers

To see that the condition Q ( a 1 ) > 0 is not sufficient, let
{ n = 0 , m = 1 a = 2 3 , b = 1 c 0 = c 1 = 1 3
Then Q(z) simplifies to
2 z 3
which has z=2 as its only root, and
Q ( a 1 ) = Q ( 3 2 ) = 1 6 > 0
Q ( a 1 ) = Q ( 3 2 ) = 1 6 > 0
and P ( z ) simplifies to
9 z 2 + 19 z 6 9 z
which has two roots, but only one of them has absolute value greater than 1

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