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Let a be a real number such that a + \frac{1}{a} + a^2 + \frac{1}{a^2} = 0.  Compute a^5.

 May 1, 2024
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Multiplying all terms by a^2 gives 

a4+a3+a+1=0(a3+1)(a+1)=0(a+1)2(a2a+1)=0a=1 or a2a+1=0

 

Note that the equation a2a+1=0 has determinant Δ=(1)24(1)(1)=3<0, so a2a+1=0 has no real solutions.

 

Hence, a = -1 is the only possibility. Then a5=(1)5=1.

 May 1, 2024

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