If p, q ,r be the roots of the equation x(1 + x^2) + x^2(6 + x) + 2 = 0 then find the value of 1/p + 1/q + 1/r

We have the equation x(1 + x^2) + x^2(6 + x) + 2 = 0

=> x + x^3 + 6x^2 + x^3 + 2 = 0

=> 2x^3 + 6x^2 + x + 2 = 0

This is a cubic equation. A cubic equation is generally denoted by the form ax^3 + bx^2 + cx + d = 0

For a cubic Equation ax^3 + bx^2 + cx + d = 0, having roots j, k ,l,  we have the following general results:-

j + k + l = -b/a

jk + kl + lj = c/a

jkl = -d/a

Returning to our problem, we have 2x^3 + 6x^2 + x + 2 = 0 which has roots p, q, r. so

p + q + r = -6/2 = -3

pq + qr + rp = 1/2

pqr = -2/2 = -1

We have to evaluate 1/p + 1/q + 1/r

= (qr + pr + pq) / pqr

= (1/2) / (-1)

= -1/2

Hence, the value of 1/p + 1/q + 1/r is -1/2