If $α,β,γ,δ$ be the roots of the equation $x^4 + px^3 + qx^2 + rx + s = 0$ ,then find in terms of $p,q,r,s$ the value of $\sum \alpha^4$
My general strategy was transforming the equation to one whose roots are $\alpha^4,etc$ but it seems to be impossible as the transformed equation with $y=\alpha^4(\implies y^{1/4}=\alpha)$ is $y+py^{3/4}+qy^{1/2}+ry^{1/4}+s=0,$ which is not even a polynomial in $y$ and hence we can't apply Vieta's formulas.
Next, I tried simplifying the expression $\sum \alpha^4$ but, it doesn't turn out favourable as a huge calculation appears which couldn't be simplified more and no desirable form was obtained and this was in vain too. I don't understand how to approach it…
Best Answer
HINT…use $$\Sigma\alpha^4=-p\Sigma\alpha^3-q\Sigma\alpha^2-r\Sigma\alpha-s\Sigma 1$$ And $$\Sigma\alpha^3=-p\Sigma\alpha^2-q\Sigma\alpha-r\Sigma 1-s\Sigma\frac{1}{\alpha}$$ And all the remaining terms can easily be obtained using Vieta etc.