Let P ( z ) = a 0 + a 1 z + ⋯ +...

Jaxon Hamilton

Jaxon Hamilton

Answered

2022-07-17

Let P ( z ) = a 0 + a 1 z + + a n z n be a polynomial whose coefficients satisfy
0 < a 0 < a 1 < < a n .

Answer & Explanation

Brenton Gay

Brenton Gay

Expert

2022-07-18Added 13 answers

The thing to do is to look instead at the polynomial
Q ( z ) = ( 1 z ) P ( z ) = ( 1 z ) ( i = 0 n a i z i ) = a 0 a n z n + 1 + i = 1 n ( a i a i 1 ) z i
Now, let | z | > 1 be a root of P ( z ), and hence a root of Q ( z ). Therefore, we have a 0 + i = 1 n ( a i a i 1 ) z i = a n z n + 1 Then, we have
| a n z n + 1 | = | a 0 + i = 1 n ( a i a i 1 ) z i | a 0 + i = 1 n ( a i a i 1 ) | z i | < a 0 | z n | + i = 1 n ( a i a i 1 ) | z n | = | a n z n |
a contradiction.

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