If z₁=2-3 i and the roots of the equation z ^3+ b z ^2+ cz + d =0 are i , z ₁ and z ₁ , then b + c + d =

Step-by-step solution

Let z₁=2-3 i, z₂=i and z_ 3^ = z ₁=2+3 i are the root of the equation (z-z₁ ) (z-z₂ ) (z-z₃ )=0 aligned & (z-i)(z-(2-3 i))(z-(2+3 i))=0 & (z-i) (z^2-4 z+13 )=0 & z^3-4 z^2+13 z-z^2 i+4 z i-13 i=0 & z^3+z^2(-4-i)+z(13+4 i)-13 i=0 aligned The given equation is z^3+b z^2+c z+d=0 ...(ii) Comparing eqn. (i) with equation (ii), we get :- b=-4-i, c=13+4 i, d=0-13 i Then, b+c+d=9-10 i .