Elementary Symmetric Polynomials, Roots of cubic polynomials I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$.  It's in a chapter about Galois Theory and is preceded by a problem about elementary symmetric polynomials.  I'm guessing I need to use elementary symmetric polynomials to solve for the two desired functions, but I really don't know where to start.  Hints would be much appreciated.  

$a$, $b$ and $c$ are the roots of $x^3 + px^2 + qx + r$ if and only if $abc = -r$, $ab+bc+ca=q$ and $-(a+b+c)=p$.
We have $\frac{1}{a_1} \cdot \frac{1}{a_2}  \cdot\frac{1}{a_3}  = \frac{1}{a_1a_2a_3}  = -1/3$, $\frac{1}{a_1}  + \frac{1}{a_2}  + \frac{1}{a_3}  = \frac{a_1a_2 + a_2a_3 + a_3a_1}{a_1a_2a_3} = \frac{-8}{-3}$ and $\frac{1}{a_1a_2}  + \frac{1}{a_2a_3}  + \frac{1}{a_3a_1}  = \frac{a_1+a_2+a_3}{a_1a_2a_3} = \frac{-7}{-3}$. It follows that $\frac{1}{a_1} $, $\frac{1}{a_2} $ and $\frac{1}{a_3} $ are the roots of $X^3 - \frac{8X^2}{3} + \frac{7X}{3} + \frac{1}{3}$. The problem with $a_1^2$, $a_2^2$, $a_3^2$ can be solved similarly.
