If $a_1,a_2,a_3$ are roots $x^3+7x^2-8x+3,$ find the polynomial with roots $a_1^2,a_2^2,a_3^2$ If $a_1,a_2,a_3$ are the roots of the cubic $x^3+7x^2-8x +3,$ find the cubic polynomial whose roots are:
$a_1^2,a_2^2,a_3^2$ and the polynomial whose roots are $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}.$ Not really sure where to go. Hints appreciated. 

Suppose $x^3+7x^2-8x+3=(x-a_1)(x-a_2)(x-a_3)$. Using difference of squares,
$$
\begin{align*}
(x^2-a_1^2)(x^2-a_2^2)(x^2-a_3^2)&=\left[(x-a_1)(x-a_2)(x-a_3)\right]\left[(x+a_1)(x+a_2)(x+a_3)\right]\\
&= \left[x^3+7x^2-8x+3\right]\left[x^3-7x^2-8x-3\right]\\
&=x^6-65 x^4+22 x^2-9.
\end{align*}
$$
Therefore $(x-a_1^2)(x-a_2^2)(x-a_3^2)=x^3-65x^2+22x-9$.
