+1839 Let a be a real number such that a + \frac{1}{a} + a^2 + \frac{1}{a^2} = 0. Find a^5.
+129852 \( a + \frac{1}{a} + a^2 + \frac{1}{a^2} = 0 \)
[a^2 + 1] / a + [ a^4 + 1] / a^2 = 0 multiply through by a^2
a^3 + a + a^4 + 1 = 0
a^4 + a^3 + a + 1 = 0
a^3 ( a + 1) + (a + 1) = 0
(a + 1) ( a^3 + 1) = 0
a + 1 = 0 or a^3 + 1 = 0
a = -1 a^3 = -1
a = -1
a^5 = (-1)^5 = -1
