polynomial can therefore be factored in 2 ways: First method of factorization This expands directly as (note the alternating signs), where We already have two solutions, and only need the other two. Hence the problem is equivalent to solving a quadratic polynomial. In particular, note that and Adding these to the above, we get and . Matching these against the coefficients from expanding , we get and From this, we easily get and . From here, we construct a quadratic equation with and apply the standard solution formula to get and Closer examination shows that: and meaning these two points are the two points on a single period-2 cycle. Second method of factorization We can factor the quartic by using polynomial long division to divide out the factors and which account for the two fixed points and (whose values were given earlier and which still remain at the fixed point after two iterations): The roots of the first factor are the two fixed points. They are repelling outside the main cardioid. The second factor has the two roots These two roots, which are the same as those found by the first method, form the period-2 orbit. Special cases Again, let us look at . Then and both of which are complex numbers. We have . Thus, both these points are "hiding" in the Julia set. Another special case is , which gives and . This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set. Cycles for period greater than 2 The degree of the equation is 2n; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8. After factoring out the factors giving the two fixed points, we would have a sixth degree equation. There is no general solution in radicals to polynomial equations of degree five or higher, so the points on a cycle of period greater than 2 must in general be computed using numerical methods. However, in the specific case of period 4 the cyclical points have lengthy expressions in radicals. In the case c = –2, trigonometric solutions exist for the periodic points of all periods. The case is equivalent to the logistic map case r = 4: Here the equivalence is given by One of the k-cycles of the logistic variable x (all of which cycles are repelling)