Prove that if p,q,r and s are odd integers, then this equation has no integer roots:
x^10 + px^9 - qx^7 + rx^4 - s = 0
Hello, the undertaker!
I baby-talked my way through it . . .
Prove that if are odd integers,
then: . .has no integer roots.
Note that:
An even integer raised to any positive integral power is even: .
An odd integer raised to any positive integral power is odd: .
We have: .
Suppose is even.
We have: .
. . . . . . . . .
And the difference of an even number and an odd number cannot be zero.
Suppose is odd.
We have: .
. . . . . . . . .
And the difference of an odd number and an even number cannot be zero.
Geez, that was awesome Soroban. Such a simple technique for such a seemingly intractable problem ! Too bad I can't triple-thank you for this one.
The only improvement I can suggest is putting the "even" and "odd" tags between \mathrm{}, like .
Otherwise,