QUESTION: The equation $x^3+px^2+qx+r=0$ has roots $\alpha$, $\beta$ and $\gamma$. Find the equation whose roots are $\alpha – \frac{1}{\beta\gamma} , \beta – \frac{1}{\alpha\gamma} , \gamma – \frac{1}{\alpha\beta} $ .
I tried to solve it by taking the sum of roots, product of roots taken two at a time, and the product of all the roots. I got some equations, could sail some far, but eventually it really messed up and now I am stuck. Can anyone help me out. Is there any other possible way to do this?
Thank you.
Best Answer
We can rewrite the expressions as $\alpha(1-\frac1{\alpha\beta\gamma})$, $\beta(1-\frac1{\alpha\beta\gamma})$, $\gamma(1-\frac1{\alpha\beta\gamma})$.
Or in other words, $\alpha(1+\frac1p)$, $\beta(1+\frac1p)$, $\gamma(1+\frac1p)$.
Let $z=1+\frac1p$. So, the sum of the expressions is $z(\alpha+\beta+\gamma)=-zr$, the sum of product pairs of the expression is $qz^2$, and the product of the expressions is $-pz^3$. So, the polynomial is $$x^3+pz^3x^2+qz^2x+rz$$