Determine if there exist rational number a and irrational number A such that A

Summer Bradford 2022-06-26 Answered
Determine if there exist rational number a and irrational number A such that A 3 + a A 2 + a A + a = 0
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pressacvt
Answered 2022-06-27 Author has 19 answers
For any integer a except 0 or 1, the polynomial x 3 + a x 2 + a x + a has no rational roots. Any rational root A would have to be an integer (by Gauss's lemma, or the Rational Root Theorem). Now A 3 + a A 2 + a A + a = 0 means
a = A 3 A 2 + A + 1 = A + 1 1 A 2 + A + 1 which, if A is an integer, is not an integer unless A = 0 (corresponding to a = 0) or A = 1 (corresponding to a = 1): otherwise A 2 + A + 1 = ( A + 1 / 2 ) 2 + 3 / 4 > 1.

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