Question

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

Answer 1

$x^3-p{x}^2+qx-r=0$

Let the roots of the equation be $a,b,c$

$a+b+c=pab+bc+ca=qabc=r$

1. )$\sum \frac{1}{a^2}=\frac{\sum a^2{b}^2}{a^2{b}^2{c}^2}=\frac{{(a+b+c)}^2-2abc(a+b+c)}{a^2{b}^2{c}^2}=\frac{p^2-2pr}{r^2}$

2. )$\sum \frac{1}{a^2{b}^2}=\frac{{(a+b+c)}^2-2(ab+bc+ac)}{a^2{b}^2{c}^2}=\frac{p^2-2q}{r^2}$