Question

If $a,b,c$ are the roots of the equation $x^3+p_1{x}^2+p_{2}x+p_3=0$, form the equation whose roots are $a^2,b^2,c^2$.

Answer 1

The required equation is $(y-a^2)(y-b^2)(y-c^2)=0$,
or $(x^2-a^2)(x^2-b^2)(x^2-c^2)=0$, if $y=x^2$;
That is, $(x-a)(x-b)(x-c)(x+a)(x+b)(x+c)=0$.
But $(x-a)(x-b)(x-c)=x^3+p_1{x}^2+p_{2}x+p_3$;
Hence $(x+a)(x+b)(x+c)=x^3-p_1{x}^2+p_{2}x-p_3$.
Thus the required equation is
$(x^3+p_1{x}^3+p_{2}x+p_3)(x^3-p_1{x}^2+p_{2}x-p_3)=0$,
or $(x^3+p_{2}x{)}^2-(p_1{x}^2+p_3{)}^2=0$,
or $x^6+(2p_2-p_1^2)x^4+(p_2^2-2p_1{p}_3)x^2-p_3^2=0$;
and if we replace $x^2$ by $y$, we obtain
$y^3+(2p_2-p_1^2)y^2+(p_2^2-2p_1{p}_3)y-p_3^2=0$.