"simple" polynomial question, any help would be appreciated
Show that (x-1)(x-2) is a factor of
P(x) = x^n (2^m -1) + x^m(1-2^n) + (2^n - 2^m)
m and n are positive integers
Well if the expressions $\displaystyle x-2$ and $\displaystyle x-1$ are factors of the given polynomial, $\displaystyle x=1$ and$\displaystyle x=2$ must be its roots.
So just substitute $\displaystyle x=1$ and then$\displaystyle x=2$ and see if the equation
$\displaystyle x^n(2^m-1) + x^m(1-2^n) + 2^n-2^m = 0$ is satisfied.
so for $\displaystyle x=1$ , the value of the polynomial is,
$\displaystyle 2^m-1+1-2^n+2^n-2^m=0$
Hence the above equation is satisfied. So x-1 is a factor.
Now, insert$\displaystyle x=2$ and prove that the value of the polynomial becomes zero.
so as x=1 and x=2 are the roots of the given polynomial $\displaystyle (x-1)(x-2)$ is a factor.