If `a,b,c,d in R` and all the three roots of `az^3 + bz^2 + cZ + d=0` have negative real parts, then

Options

A. `ab gt 0 `
B. `bv gt 0`
C. `ad gt 0`
D. `bc-ad gt 0`

Answer

A, B, C, D

Explanation

Let the roots of `az^(3) + bz^(2) + cz + d = 0 ` be `z_(1) = x_(1),z_(2) = x_(2) + iy_(2)` and `z_(3) = x_(2) -iy_(2)`
`therefore z_(1) + z_(2) + z_(3) = -(b)/(a)`
`rArr x_(1) +2x_(2) = - (b)/(a) lt 0`
`rArr ab gt 0`
`z_(1)z_(2) + z_(2)z_(3) + z_(1) +z_(3) = (c)/(a)`
`x_(1)(x_(2) + iy_(2)) + (x_(2)^(2) + y_(2)^(2)) + x_(1)(x_(2) -iy_(2))`
` =2x_(1)x_(2) + x_(2)^(2) + y_(2)^(2) gt 0`
`rArr (c)/(0) gt0`
`rArr ac gt 0`
`rArr a^(2) bc gt 0 `
`bc gt 0`
Also, `z_(1)z_(2)z_(3)= x_(1)(x_(2)^(2) + y_(2)^(2)) = -(d)/(a)`
`rArr ad gt 0`
Futher,`-(bc)/(a^(2)) = (-(b)/(a))((c)/(a))`
`= (x_(1)+ 2x_(2)) (2x_(1)x_(2) + x_(2)^(2) + y_(2)^(2))`
`=x_(1)(x_(2)^(2) + y^(2)_(2)) + 2x_(1)^(2) x_(2) + 2x_(1)x_(2)^(2) + 2x_(2) (x_(2)^(2) + y_(2)^(2))`
`lt x_(1)(x_(2)^(2) + y_(2)^(2))`
`rArr -(bc)/(a^(2)) lt - (d)/(a)`
`rArr bc gt ad`