Math Help - No integer solutions to x^10 + px^9 - qx^7 + rx^4 - s = 0

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,