need some help pleasee
thank you very much
Prove that if p,q,r and s are odd integers, then
x^10 + px^9 - qx^7 + rx^4 - s = 0
has no integer roots
thank you in advance
Hello, ieatfood!
Prove that if are odd integers,
then: has no integer roots.
[1] The sum of the ten roots is , an odd integer.
[2] The product of the ten roots is , an odd integer.
From [2] . . . Since is odd, all ten roots must be odd integers.
But the sum of ten odd integers is even . . . which contradicts [1].
If you take the polynomial and multiply them out you get as coefficients (without signs) the so-called "symettric functions":
(all possible products)
...
The important thing about these is that they are invariant under and premutation of the roots.
This is the (unsigned) version of the theorem Soroban used.