2011/098) if a=0.5(5-root 21) then find a^3 + a^-3- 5a^2- 5a^-2+ a+ a^-1.?

a = .5(5-sqrt(21)

so 1/a = 2/(5- sqrt(21) = 2(5+ sqrt(21)/(5^2-21) = .5(5 + sqrt(21))

so a + 1/a = 5

a^3+ 1/a^3- 5 ( a^2+ 1/a^2) + (a+ 1/a)
= (a+1/a)^3 - 3 ( a + 1/a) - 5((a+1/a2^2) - 2) + (a+ 1/a)
= 5^3 - 3 * 5 - 5(5^2 - 2) + 5
= 125 - 15 - 125 + 10 + 5 = 0