Find k such that f(x) = x^4 - kx^2 + kx+2 has the factor x -

Question

please help, I can’t find the answer

Answer 1

If we let k=2^(2/3) then we obtain that f(x)=x^2(x^2-k)+kx+2=x^2(x-sqrt(k))(x+sqrt(k))+kx+2 and we see by our choice of k we actually have x+sqrt(k)=x+2^(1/3)=(1/2^(2/3))(2^(2/3)x+2). Hence, we may rewrite the function f(x) as x^2(x-2^(1/3))(x+2^(1/3))+2^(2/3)(x+2^(1/3))=[x^2(x-2^(1/3))+2^(2/3)] (x+2^(1/3)) which means that f(x) has a factor of the form x - something where something=-2^(1/3).