Grouping to Solve

When a cubic has four terms and no factor shared by all of them, try grouping. Take x^3 - 3x^2 - 9x + 27 = 0. The first two terms share x^2 and the last two share -9, so the left side is x^2(x - 3) - 9(x - 3). Now x - 3 is a common factor: (x - 3)(x^2 - 9) = 0. The bracket x^2 - 9 is a difference of squares, so the whole equation is (x - 3)(x + 3)(x - 3) = 0.

Splitting into cases gives 3, then -3, then 3 again. The factor x - 3 appears twice, so 3 is a repeated root. In the answer it counts once: the solutions are 3 and -3. An answer is a set of numbers, and a set holds each number once, however many factors point to it. On a graph, a root repeated an even number of times, like 3 here, is where the curve touches the x-axis and turns back. A root repeated an odd number of times crosses: (x - 1)^3 crosses at 1.