Find the binomial factors for: x=x^4-6x^3+11x^2-6x

If there is a linear factor with rational coefficients it is either (x+1) or (x-1) because the coefficients must be integers and the constant term divides 1.

This means that 1 or - 1 is a root. Neither fits so we can't find a linear factor.

So try (x^2 + kx + 1) (x^2 + cx + 1) or (x^2 + kx - 1) (x^2 + cx - 1).

In the first case the coefficients of x ^3 and x are both k+c and the coefficient of x^2 is 2 + kc. I. e. k+c = 6 and kc = 5. This has a solution k = 1, c = 5.

This gives the answer. The second possibility with - 1s gives k+c = both 6 and - 6 so doesn't work.