If a is a root of x 2 </msup> ( x &#x2212;<!-- − --> 5 ) + 2 , then what

boloman0z

boloman0z

Answered question

2022-06-24

If a is a root of x 2 ( x 5 ) + 2, then what is [ a 4 ] ?

Answer & Explanation

Cristian Hamilton

Cristian Hamilton

Beginner2022-06-25Added 23 answers

Let a,b,c be the zeros with a the largest. By expanding we see that
f ( x ) f ( x ) = 4 20 x 2 + 25 x 4 x 6 = g ( x 2 ) ,
where
g ( x ) = x 3 25 x 2 + 20 x 4
has a 2 , b 2 and c 2 as its zeros. By the Vieta relations it follows that
s 1 := a 2 + b 2 + c 2 = 25
and the second symmetric polynomial
s 2 := a 2 b 2 + a 2 c 2 + b 2 c 2 = 20.
Therefore
a 4 + b 4 + c 4 = s 1 2 2 s 2 = 585.
a 4 + b 4 + c 4 = s 1 2 2 s 2 = 585.
But f ( 3 / 4 ) < 0 < f ( 0 ) and f ( 3 / 4 ) < 0, so | b | and | c | are both < 3 / 4. Therefore 0 < b 4 + c 4 < 1 and hence
584 < a 4 < 585.

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