Matematické Fórum

Skrytý text:

If a,b and c are odd then we can write them as:
where
If the given polynomial should have only rational roots then the discriminant of that polynomial is either zero or a square number. So lets look at the discriminant:

We can see it can never be a zero given that for every , because all of them are integers and there are no fractions etc. so let call , where . Then we have the equation:
, where so that the discriminant of that polynomial is a square number.
Then we get:

The left side of that equation is an integer so have to be the right side. We can see that the right side is an integer for every , where so in other words for every odd number. Then we can express from that equation:
and put it back into . So we get:

that is the same as:

The left side is an even number for every and the right side is an odd number for every so the equation is never right -> this polynomial cannot have racional roots. Q.E.D

What I am not sure about: does have to be an integer so that is an integer?