1 Let P ( z )= z 3+ az z 2+ bz + c where a , b and c are real. There exists a complex number ω such that the three roots of P(z) are ω+3 i, ω+9 i and 2 ω-4 where i2=-1. Find the value of mid a+b+ c mid.

Question

1 Let P ( z )= z 3+ az z 2+ bz + c where a , b and c are real. There exists a complex number ω such that the three roots of P(z) are ω+3 i, ω+9 i and 2 ω-4 where i2=-1. Find the value of mid a+b+ c mid. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

It is to be noted that two of the numbers need to be conjugates and one number must be real, as the coefficients of the cubic are all real.
Three roots are,

ω + 3 i, ω + 9 i

and

2 ω - 4

let

ω + 3 i

is real, hence

ω = α - 3 i

where

α \in R

then

(α - 3 i + 9 i)

and

(2 α - 6 i - 4)

i.e.

α + 6 i

and

(2 α - 4) - 6 i

must be complex conjugate

\Rightarrow α = 2 α - 4

∴ α = 4

Hence the roots are
4,4+6i, 4-6i (other options not possible)
the equation is

(z - 4) (z^{2} - 8 z + 5 z) \Rightarrow z^{3} - 12 z^{2} + 84 z - 208

Hence

∣ a + b + c ∣ = ∣ - 12 + 84 - 208∣ = 136

.

Q. 1 Let P(z)=z3+azz2+bz+c where a,b and c are real. There exists a complex number ω such that the three roots of P(z) are ω+3i,ω+9i and 2ω−4 where i2=−1. Find the value of ∣a+b+ c∣.

Q. 1 Let $P (z) = z^{3} + a z z^{2} + b z + c$ where $a, b$ and $c$ are real. There exists a complex number $ω$ such that the three roots of $P (z)$ are $ω + 3 i, ω + 9 i$ and $2 ω - 4$ where $i^{2} = - 1$ . Find the value of $∣ a + b +$ $c ∣$ .