MHB Roots of p ( z ) in IR [ z ] (Lava's question at Yahoo Answers)

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The discussion focuses on solving the polynomial equation z^3 + az^2 + bz + c = 0, where a, b, and c are real numbers and the equation has a purely imaginary root. It establishes that if βi is a root, then -βi must also be a root due to the polynomial's real coefficients. The presence of a real root α leads to the conclusion that the roots can be expressed as z1 = α, z2 = βi, and z3 = -βi. Using Cardano-Vieta relations, the relationships between the coefficients and the roots are derived, resulting in the conditions α = -a, β^2 = b, and αβ^2 = -c. Ultimately, it concludes that the roots of p(z) can be represented as z1 = -a, z2 = +√|b|i, and z3 = -√|b|i.
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Hello Lava,

Denote $z_1,z_2,z_3$ the roots of $p(z)=z^3+az^2+bz+c\in\mathbb{R}[z]$. If $\beta i$ is a root of $p(z)$, then $-\beta i$ is also a root of $p(z)$ ($p(z)$ is a real polynomial). By hypothesis $\beta\ne 0$ and $p(z)$ has a real root $\alpha$ (odd degree). If $z_1=\alpha$, $z_2=\beta i$, $z_3=-\beta i$, and using the Cardano-Vieta relations: $$\left \{ \begin{matrix} z_1+z_2+z_3=-a\\z_1z_2+z_1z_3+z_2z_3=b\\z_1z_2z_3=-c\end{matrix}\right.\Leftrightarrow \left \{ \begin{matrix} \alpha=-a\\\beta^2=b\\\alpha\beta^2=-c\end{matrix}\right.$$ As $\beta^2=b=c/a$, necessarily $c=ba.$ So the roots of $p(z)$ are $z_1=-a$, $z_2=+\sqrt{|b|}i$, $z_3=-\sqrt{|b|}i$.
 
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