I am at a loss at how to approach this problem. It doesn't make sense to me to solve without finding the roots. I couldn't find the roots without a calculator anyway, but you're not meant to.
Show that if the roots of the equation:
$5x^3-x^2-2x+3=0$
are a,b and c then:
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2/3$
$a^2+b^2+c^2=\frac{21}{25}$
$a^3+b^3+c^3=-\frac{194}{125}$