Solve the equation #1000-x^3 = 0#. There are three roots, one of which is #x = 10#, the other two of are non-real; denote them as #alpha# and #beta#.
When factorized, the expression will look like:
#1000-x^3 = -(x-10)(x-alpha)(x-beta)#
Even though #alpha# and #beta# are non-real, the expression #(x-alpha)(x-beta)# will be a quadratic expression with real coefficients. To show this, divide both expression by #-(x-10)# and perform long division.
#(x - alpha)(x - beta) = -frac{1000 - x^3}{x-10}#
#= frac{x^3 - 1000}{x - 10}#
#= frac{x^3 color(green)(- 10x^2)}{x - 10} + frac{color(green)(10x^2) - 1000}{x - 10}#
#= x^2 + frac{10x^2 color(blue)(- 100x)}{x - 10} + frac{color(blue)(100x) - 1000}{x - 10}#
#= x^2 + 10x + 100#
Therefore, #1000 - x^3 = - (x - 10)(x^2 + 10x + 100)#
Note: It may be useful to memorize #a^3 - b^3 = (a - b)(a^2 + ab + b^2)#
