Question

If $a,b,c$ are the roots of the equation $x^3-p{x}^2+qx-r=0$ find the value of $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$.

Answer 1

$a,b,c$ are the roots of the equation $x^3-p{x}^2+qx-r=0$

Write $\frac{1}{y}$ for $x$, multiply by $y^3$, and change all the signs;

Then the resulting equation is $r{y}^3-q{y}^2+py-1=0$
And it its roots $\frac{1}{a},\frac{1}{b},\frac{1}{c}$;
Hence $\sum \frac{1}{a}=\frac{q}{r},\sum \frac{1}{ab}=\frac{p}{r}$;
$\therefore \sum \frac{1}{a^2}={(\sum \frac{1}{a})}^2-2(\sum \frac{1}{ab})=\frac{q^2-2pr}{r^2}$.