Show that the equation `Z^4+2Z^3+3Z^2+4Z+5=0` has no root which is either purely real or purely imaginary.

Explanation

`Z^(4) + 2Z^(3)+ 3Z^(2) + 4Z + 5=0`
If `alpha` is a purely real root of the above equation, then
`alpha^(4) + 2alpha^(3) + 3alpha ^(2) + 4alpha + 5 =0`
`therefore " " (alpha^(2) + alpha)^(2) +2(alpha + 1)^(2) + 3 = 0`which of is not possible as all
the terms of LHS are positive .
If i`beta` is a purely imaginary root of the given equation, then
`beta^(4) - 2ibeta -3beta^(2) + 4ibeta + 5=0`
`therefore bet^(4) -3beta^(2) + 5 = 0 and 2beta^(3) - 4beta = 0`
i.e., `(beta^(2) - (3)/(2))^(2) +(11)/(4) = 0` and `2beta(beta^(2) -2) =0`
So,no value of `beta` exists.
Hence, equation has neither purely real nor purely imaginary roots.