In $\mathbb{F}_2[x]$ check the polynomial $x^7+x^4+x^2+x+1$ for multiple factors.

Question

In $\mathbb{F}_2[x]$ check the polynomial $x^7+x^4+x^2+x+1$ for multiple factors.

Let $f(x)=x^7+x^4+x^2+x+1$. Then $f'(x)=7x^6+4x^3+2x+1 = x^6+1$.

Then using Euclid's algorithm I calculated $\gcd(f,f')$:

$x^7+x^4+x^2+x+1=(x^6+1)x+(x^4+x^2+1)$

$x^6+1 = (x^4+x^2+1)(x^2+1) + 0$

Hence $\gcd(f,f')=x^4+x^2+1$, so $f$ and $f'$ are not coprime, and therefore $f$ will have multiple factors.

Is this correct?