Question

How do you factor `1000-x^3`?

Answer 1

`(10-x)(100+10x+x^2)`

Explanation:

This is a difference of cubes.

The general form for a difference of cubes is

`a^3-b^3=(a-b)(a^2+ab+b^2)`

In your case,

`1000-x^3=(10)^3-(x)^3`

so

`a=10`
`b=x`

Thus,

`1000-x^3=(10-x)(100+10x+x^2)`

Answer 2

`1000-x^3=(10-x)(100+10x+x^2)`

Explanation:

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)`