Transformation of Roots for a Cubic Polynomial

Question

Consider $$x^3 + px + q = 0$$ with roots $ \alpha, \beta. \gamma$

We want to find the degree 3 polynomial that has roots : $$ \frac{\alpha\beta}{\gamma}, \frac{ \alpha\gamma}{\beta}, \frac{ \beta\gamma }{\alpha} $$

My attempt so far:

$$\frac{ \alpha\beta}{\gamma} = \frac{\alpha\beta\gamma}{\gamma^2} = - \frac{q}{\gamma^2} $$ and similarly for all of the others:

So now I let $$ t = - \frac{q}{x^2} $$ and substitute into the original polynomial; except I will multiply everything by $x$ first to make my substitution easier.

$$ x^4 + px^2 = -qx $$ $$ \frac{q^2}{t^2} - \frac{pq}{t} = - q\sqrt{ -\frac{q}{t} } $$ Squaring both sides and multiplying by $t^4$, we obtain: $$qt^3 +p^2t^2 -2pqt +q^2 = 0 $$

Except this seems like a messy, tedious solution as you have to muck around with square roots etc. Is there a nicer way to do this?

Answer 1

You just have to express the elementary symmetric functions of $\;\dfrac{\alpha\beta}\gamma, \dfrac{\beta\gamma}\alpha$ and $\dfrac{\gamma\alpha}{\beta}$ with $p, q,r$, then use Vieta's relations.

• $\begin{aligned}[t]\dfrac{\alpha\beta}\gamma+ \dfrac{\beta\gamma}\alpha+\dfrac{\gamma\alpha}{\beta}&=\dfrac{(\alpha\beta)^2+(\beta\gamma)^2+(\gamma\alpha)^2}{\alpha\beta\gamma}=\dfrac{(\alpha\beta+\beta\gamma+\gamma\alpha)^2-2(\alpha^2\beta\gamma+\alpha\beta^2\gamma+\alpha\beta\gamma^2)}{\alpha\beta\gamma}\\&=\dfrac{(\alpha\beta+\beta\gamma+\gamma\alpha)^2}{\alpha\beta\gamma}-2(\alpha+\beta+\gamma)=\frac{p^2}{-r}\end{aligned} $
• $\dfrac{\alpha\beta}\gamma\cdot\dfrac{\beta\gamma}\alpha+\dfrac{\beta\gamma}\alpha\cdot\dfrac{\gamma\alpha}{\beta}+\dfrac{\gamma\alpha}{\beta}\cdot\dfrac{\alpha\beta}\gamma=\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)=-2p$
• $\dfrac{\alpha\beta}\gamma\cdot\dfrac{\beta\gamma}\alpha\cdot\dfrac{\gamma\alpha}{\beta}=\alpha\beta\gamma=-r$

Hence the equation is $$x^3+\frac{p^2}r x^2-2px+r=0\iff rx^3+p^2x^2-2prx+r^2=0.$$

Answer 2

Here is a hint for a way of solving it using elementary steps: Consider a degree 3 polynomial that have roots $\alpha, \beta, \gamma$:

$$ (x-\alpha)(x-\beta)(x-\gamma), $$ this must then be the same polynomial as $$ x^3+px+q. $$

Can you now relate now $\alpha, \beta, \gamma$ with $p$ and $q$? Do we get any relation between $\alpha, \beta, \gamma$?

Yes, from $$ (x-\alpha)(x-\beta)(x-\gamma)=x^3+px+q $$ one obtains $\alpha \beta + \alpha \gamma + \beta\gamma= p$, $\alpha \beta \gamma =-q$, and $\alpha + \beta + \gamma=0$. Due to this latter relation, one can also write $(\alpha + \beta + \gamma)^2=0$, i.e., $\alpha^2 +\beta^2 +\gamma^2=-2(\alpha \beta + \alpha \gamma + \beta\gamma)=-2p$ and also $(\alpha \beta)^2 + (\alpha \gamma)^2 + (\beta \gamma)^2=(\alpha \beta + \alpha \gamma + \beta \gamma)^2-2(\alpha \beta\gamma)(\alpha +\beta+\gamma)=(\alpha \beta + \alpha \gamma + \beta \gamma)^2=p^2$

After this, consider a degree 3 function that has roots $\frac{\alpha \beta}{\gamma}, \frac{\alpha \gamma}{\beta}, \frac{\alpha \gamma}{\beta}$:

$$ \left(x-\frac{\alpha \beta}{\gamma}\right)\left(x-\frac{\alpha \gamma}{\beta}\right)\left(x-\frac{\alpha \gamma}{\beta}\right)=\\ \frac{1}{\alpha \beta \gamma} \left[(\alpha \beta \gamma) x^3 - ((\alpha \beta)^2 + (\alpha \gamma)^2+(\beta \gamma)^2)x^2 + (\alpha \beta \gamma)(\alpha^2 + \beta^2 +\gamma^2)x -(\alpha\beta\gamma)^2\right] $$

Now use the obtained identities between $\alpha, \beta, \gamma$ and $p, q$ to write up the above degree 3 polynomial with the desired roots:

$$ qx^3+p^2x^2-2pqx +q^2. $$