Solve the equation 1000-x^3 = 01000−x3=0. There are three roots, one of which is x = 10x=10, the other two of are non-real; denote them as alphaα and betaβ.
When factorized, the expression will look like:
1000-x^3 = -(x-10)(x-alpha)(x-beta)1000−x3=−(x−10)(x−α)(x−β)
Even though alphaα and betaβ are non-real, the expression (x-alpha)(x-beta)(x−α)(x−β) will be a quadratic expression with real coefficients. To show this, divide both expression by -(x-10)−(x−10) and perform long division.
(x - alpha)(x - beta) = -frac{1000 - x^3}{x-10}(x−α)(x−β)=−1000−x3x−10
= frac{x^3 - 1000}{x - 10}=x3−1000x−10
= frac{x^3 color(green)(- 10x^2)}{x - 10} + frac{color(green)(10x^2) - 1000}{x - 10}=x3−10x2x−10+10x2−1000x−10
= x^2 + frac{10x^2 color(blue)(- 100x)}{x - 10} + frac{color(blue)(100x) - 1000}{x - 10}=x2+10x2−100xx−10+100x−1000x−10
= x^2 + 10x + 100=x2+10x+100
Therefore, 1000 - x^3 = - (x - 10)(x^2 + 10x + 100)1000−x3=−(x−10)(x2+10x+100)
Note: It may be useful to memorize a^3 - b^3 = (a - b)(a^2 + ab + b^2)a3−b3=(a−b)(a2+ab+b2)
