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. `bc gt 0`
C. `ad gt 0`
D. `bc-ad gt 0`

Answer

A, B, C

Explanation

`(a,b,c)` Let `z_(1)=x_(1)`, `z_(2)`, `z_(3)=x_(2)+-iy_(2)`
`implies z_(1)+z_(2)+z_(3)=-(b)/(a)`
`impliesx_(1)+2x_(2)=-(b)/(a) lt 0impliesab gt 0`
Also, `z_(1)z_(2)z_(3)=x_(1)[x_(2)^(2)+y_(2)^(2)]=-(d)/(a)lt0impliesad gt 0`
Also `z_(1)z_(2)+z_(2)z_(3)+z_(1)z_(3)=(c )/(a)`
`implies x_(1)(x_(2)+iy_(2))+x_(1)(x_(2)-iy_(2))+x_(2)^(2)+y_(2)^(2)=2x_(1)x_(2)+x_(2)^(2)+y_(2)^(2) gt 0`
`implies (c )/(a) gt 0`
`implies (b)/(a) (c )/(a) gt 0impliesbc gt 0`