<s>[INST] How to find GCD of those two complex polynomials? I have a polynomial
$$f(x) = i(x^2-1)^3+(x^2+1)^3-8x^3$$
I want to check if this has repeated roots. To do so, I'll find greatest common divisor (euclidean algorithm) of $f(x)$ and its derivative $f'(x)$. 
$$f(x) = i(x^2-1)^3+(x^2+1)^3-8x^3 = i(x^6-3x^4+3x^2-1)+(x^6+3x^4+3x^2+1)-8x^3$$
$$f'(x) = i(6x^5-12x^3+6x)+(6x^5+12x^3+6x)-24x^2$$
I know I should find the GCD as $(x-i)(x-1)$ but this is where I'm stuck. What would the next step be? 
 [/INST] By inspection, $f(1)=0$ so $x-1$ is a factor of $f(x)$.
Since $f'(x)= 3i(x^2-1)^2(2x)+3(x^2+1)^2(2x)-24x^2$, we also have that $f'(1)=0$, so $x-1$ is a factor of $f'(x)$.
This means $(x-1)^2$ is a factor of $f(x)$.

One also has that by inspection $f(i)=0$ so $x-i$ is a factor of $f(x)$.
We also have that $f'(i)=0$, so $x-i$ is a factor of $f'(x)$.
So we also have that $(x-i)^2$ is a factor of $f(x)$.
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