Condition for Quartic with Real Coefficients to have Wholly Imaginary Root
Theorem
Let $Q$ be the quartic equation:
- $(1): \quad z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$
such that all of $a_1, a_2, a_3, a_4$ are real numbers.
Then $Q$ has a root which is wholly imaginary if and only if:
- $\text {(a)}: \quad a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$
- $\text {(b)}: \quad a_1 a_3 > 0$
Proof
Necessary Condition
We have:
\(\ds a_3^2 + a_1^2 a_4\) | \(=\) | \(\ds a_1 a_2 a_3\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_4\) | \(=\) | \(\ds \frac {a_3} {a_1} \paren {a_2 - \frac {a_3} {a_1} }\) |
This leads to the factorisation of $(1)$:
\(\ds z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {z^2 + \frac {a_3} {a_1} } \paren {z^2 + a_1 z + \paren {a_2 - \frac {a_3} {a_1} } }\) | \(=\) | \(\ds 0\) |
Two solutions to this are found from:
\(\ds z^2 + \frac {a_3} {a_1}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^2\) | \(=\) | \(\ds -\frac {a_3} {a_1}\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \pm \sqrt {-\frac {a_3} {a_1} }\) |
The other two solutions are found by solving:
\(\ds z^2 + a_1 z + \paren {a_2 - \frac {a_3} {a_1} }\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds -\frac {a_1} 2 \pm \frac {\sqrt {a_1^2 - 4 \paren {a_2 - a_3 / a_1} } } 2\) | Quadratic Formula |
As $a_1 \ne 0$ it follows that $(3)$ always has a real part.
So only the $(2)$ can produce roots which are wholly imaginary, and that can happen only when:
- $a_3 / a_1 > 0$
that is:
- $a_1 a_3 > 0$
Thus, if $a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$ and $a_1 a_3 > 0$, $Q$ has wholly imaginary roots.
Sufficient Condition
Let $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ have wholly imaginary roots.
Then it can be written:
- $\paren {z^2 + b^2} \paren {z^2 + m z + n} = 0$
By long division of $Q$ by $z^2 + b^2$:
- $\paren {z^2 + b^2} \paren {z^2 + a_1 z + \paren {a_2 + b^2} } + \paren {a_3 - a_1 b^2} z + a_4 - \paren {a_2 - b^2} b^2 = 0$
For this to be in the above format, it is necessary that:
- $\paren {a_3 - a_1 b^2} z + a_4 - \paren {a_2 - b^2} b^2 \equiv 0$
That is:
\(\text {(4)}: \quad\) | \(\ds a_3 - a_1 b^2\) | \(=\) | \(\ds 0\) | |||||||||||
\(\text {(5)}: \quad\) | \(\ds a_4 - \paren {a_2 - b^2} b^2\) | \(=\) | \(\ds 0\) |
Hence:
\(\ds b^4 - a_2 b^2 + a^4\) | \(=\) | \(\ds 0\) | from $(4)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds b^2\) | \(=\) | \(\ds \frac {a_2 \pm \sqrt {a_2^2 - 4 a^4} } 2\) | Quadratic Formula | ||||||||||
\(\text {(6)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \paren {2 b^2 - a_2}^2\) | \(=\) | \(\ds a_2^2 - 4 a^4\) | rearranging and squaring both sides | |||||||||
\(\ds b^2\) | \(=\) | \(\ds \frac {a_3} {a_1}\) | from $(5)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\frac {2 a_3} {a_1} - a_2}^2\) | \(=\) | \(\ds a_2^2 - 4 a^4\) | substituting for $b^2$ into $(6)$, rearranging and squaring both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\frac {2 a_3} {a_1} - a_2}^2\) | \(=\) | \(\ds a_2^2 - 4 a^4\) | substituting for $b^2$ into $(6)$, rearranging and squaring both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 \frac {a_3^2} {a_1^2} + a_2^2 - \frac {4 a_2 a_3} {a_1}\) | \(=\) | \(\ds a_2^2 - 4 a_4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_3^2 + a_1^2 a_4\) | \(=\) | \(\ds a_1 a_2 a_3\) |
Hence if $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ has wholly imaginary roots, then:
- $a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$
We also see that the roots themselves are $\pm b$, where:
- $b = \sqrt {-\dfrac {a_3} {a_1}}$
Thus we have also shown that $a_3 a_1 > 0$ in order for those roots to indeed be wholly imaginary as required.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $129$
- (although see Condition for Quartic with Real Coefficients to have Wholly Imaginary Root/Mistake for analysis of an error in that work)