The discussion revolves around proving the inequality \(x^4+x^3y+x^2y^2+xy^3+y^4 > 0\) for positive values of \(x\) and \(y\). Participants are exploring various approaches to establish this inequality, with a focus on algebraic manipulation and factorization.
The conversation is ongoing, with various hints and suggestions being offered. Some participants have made progress in their understanding, while others are still seeking clarity on specific points. There is acknowledgment of special cases, particularly when \(x\) and \(y\) are equal.
Participants note that both \(x\) and \(y\) must be greater than zero, and there is discussion about the implications of \(x\) and \(y\) being equal, which raises questions about the validity of certain simplifications.
That'll do it. So how can you use that to rewrite the original expression?silina01 said:x^n - y^n = (x - y) (x^n-1 + x^n-2y+ ...+ xy^n-2 + y^n-1 )??
haruspex said:That'll do it. So how can you use that to rewrite the original expression?
Right, but there is one special case you need to address separately.silina01 said:ahhhh I see it know, in both cases if x-y <0 or if x-y>0 the quotient will always be positive. Thanks everyone.
Your expression involves 1/(x-y). What doubt should that create?silina01 said:what would that be?
silina01 said:x and y cannot be 0 by the way
Dick said:But they can be nonzero and equal.
dirk_mec1 said:But if you simplify the fraction then there is no problem.