<s>[INST] Find the cubic equation of roots $α, β, γ$. Taken from Fitzpatrick $4$ unit course textbook. The question says:
If the cubic equation $\ ax^3+bx^2+cx+d$ has roots $α, β, γ$. Find the cubic equation who's roots are $α^2, β^2, γ^2$
I keep getting a $±$ sign that I can't get rid of. The answer in the back is $x(ax+c)^2=(bx+d)^2$
Thanks for any help.
 [/INST] Hint We can write the given cubic polynomial as
$$ax^3 + bx^2 + cx + d = a(x - \alpha) (x - \beta) (x - \gamma).$$
Expanding the r.h.s. and comparing like terms gives
\begin{align}
b &= -a(\alpha + \beta + \gamma) \\
  &\,\,\vdots
\end{align}
(These are essentially Vieta's Formulas.)

Additional hint Similarly, the cubic polynomial $A X^3 + B X^2 + C X + D$ whose roots are $\alpha^2, \beta^2, \gamma^2$ can be written as $$A X^3 + B X^2 + C X + D = (x - \alpha^2) (x - \beta^2) (x - \gamma^2),$$ and again expanding the r.h.s. gives, e.g., \begin{align} B &= -(\alpha^2 + \beta^2 + \gamma^2) \\ &\,\,\vdots \end{align} (Of course, this polynomial is only determined up to multiplication by a nonzero constant, which in particular does not change the roots. The choice of $1$ for the leading coefficient above simplifies some computations.)  All that remains now is to write the coefficients $B, C, D$ in terms of the coefficients $a, b, c, d$. The two applications of Vieta's Formulas above expressions all of these coefficients in terms of symmetric functions of the roots $\alpha, \beta, \gamma$, so the coefficients can be related in terms of Newton's Identities, which give, e.g., $$\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\beta \gamma + \gamma \alpha + \alpha \beta).$$ Rearranging earlier equations and substituting give e.g., $$-B = \left(-\frac{b}{a}\right)^2 - 2 \left(\frac{c}{a}\right),$$ or $$B = \frac{2 a c - b^2}{a^2}.$$

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