Question

If alpha , beta and gamma b the roots of the equation x(1+x²)+x²(6+x)+2=0 then the value of alpha raised to minus 1 + beta raised to minus 1+ gamma raised to minus 1=?

Answer 1

Final Answer : -1/2
Let the roots of equation be a, b, c.
Steps:
1)

The given equation becomes
$2 {x}^{3} + 6 {x}^{2} + x + 2 = 0$

We know that,
a + b + c = -6/2 = -3
ab+bc+ca=1/2
abc= -2/2= -1
Now,
asked : a^(-1) + b^(-1) + c^(-1) = 1/a + 1/b + 1/c
= ab+bc+ca/(abc)
= 1/2 /(-1) = -1/2


So, Final answer is -1/2