I am currently considering a family of polynomials in $t$, defined as follows: $$P_n(t)=nt^{2n-1}-nyt^{n-1}+t-x$$Here, $x$ and $y$ are constants. It is of interest when this polynomial has a multiple root - when its discriminant is equal to zero. For $P_2(t)$, the expression is $2(2y-1)^3-27x^2$, and for $P_3(t)$, the expression is the more complicated $8748xy^5-729y^4+20250x^2y^2-4800xy+9375x^4+256$.
I am now stuck; I cannot find an expression for $P_4(t)=4t^7-4yt^3+t-x$, or one in general. Is there a simple form? Also, does anyone happen to be able to find an expression for $P_4$? If that can't be done, is it possible to know the highest degree of $y$ that will be found in this discriminant expression?
Thanks a lot - any help is much appreciated.