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

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## Source Work

1964: Murray R. Spiegel: *Theory and Problems of Complex Variables*

- 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 $a_1 a_3 > 0$.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $129$