If " $2 i$ " is a root of $f(z)=z^4+z^3+2 z^2+4 z-8=0$, then which among the following cannot be a root of…
If " $2 i$ " is a root of $f(z)=z^4+z^3+2 z^2+4 z-8=0$, then which among the following cannot be a root of $f(z)=0$ ?
$-2 i$
1
–2
2
Solution
It is given that, $f(z)=z^4+z^3+2 z^2+4 z-8$ have a root $2 i$, so one more root will be $-2 i$, so $\left(z^2+4\right)$ is the factor of $z^4+z^3+2 z^2+4 z-8$.
So, $z^4+z^3+2 z^2+4 z-8$
$
=\left(z^2+4\right)\left(z^2+z-2\right)
$
and $z^2+z-2=(z+2)(z-1)$
Therefore, the roots of $f(z)$ are $2 i,-2 i,-2$ and 1