Let `r_1, r_2, r_3` be the three (not necessarily distinct) solution to the equation `x^3 + 4x^2 - ax +1 = 0`. If a can be any real number, then find the minimum value of `(r_1+1/r_1)^2+(r_2+1/r_2)^2+(r_3+1/r_3)^2`.

Options

A. even
B. odd
C. prime
D. divisible by `'13'`

Answer

B, C, D

Explanation

`r_(1) + r_(2) + r_(3) = -4, r_(1)r_(2) + r_(2)r_(3) + r_(3)r_(1) = -a, r_(1) r_(2)r_(3) = -1`
`E = sumr_(1)^(2) + sum(1)/(r_(1)^(2))+6`
`=(sumr_(1))^(2) - 2 sumr_(1)r_(2) + [((r_(2)r_(3))^(2) + (r_(3)r_(1))^(2) + (r_(1)r_(2)))/(1)] + that`
`= 22 + 2a + [(sumr_(1)r_(2))^(2) - 2(r_(1)^(2)r_(2)r_(3) + r_(2)^(2)r_(1)r_(3) + r_(3)^(2)r_(1)r_(2))]`
`:. 22 + 2a + [a^(2) + 2(-4)] = 22 - 8 + a^(2) + 2a = (a + 1)^(2) + 13`
`min = 13` at `a = -1`