Condition for Quartic with Real Coefficients to have Wholly Imaginary Root/Mistake

Source Work

 * Chapter $1$: Complex Numbers
 * Supplementary Problems: $129$

This mistake can be seen in the 1981 printing of the second edition (1974) as published by Schaum: ISBN 0-070-84382-1

Mistake

 * $\text{(a)} \quad$ Show that the equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ where $a_1, a_2, a_3, a_4$ are real constants different from zero, has a
 * pure imaginary root if ${a_3}^2 + {a_1}^2 a_4 = a_1 a_2 a_3$.

As demonstrated in Condition for Quartic with Real Coefficients to have Wholly Imaginary Root it is also necessary for $a_1 a_3 > 0$.