# How do you factor 1000-x^3?

Dec 17, 2015

$\left(10 - x\right) \left(100 + 10 x + {x}^{2}\right)$

#### Explanation:

This is a difference of cubes.

The general form for a difference of cubes is

${a}^{3} - {b}^{3} = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$

$1000 - {x}^{3} = {\left(10\right)}^{3} - {\left(x\right)}^{3}$

so

$a = 10$
$b = x$

Thus,

$1000 - {x}^{3} = \left(10 - x\right) \left(100 + 10 x + {x}^{2}\right)$

Dec 17, 2015

$1000 - {x}^{3} = \left(10 - x\right) \left(100 + 10 x + {x}^{2}\right)$

#### 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:

$1000 - {x}^{3} = - \left(x - 10\right) \left(x - \alpha\right) \left(x - \beta\right)$

Even though $\alpha$ and $\beta$ are non-real, the expression $\left(x - \alpha\right) \left(x - \beta\right)$ will be a quadratic expression with real coefficients. To show this, divide both expression by $- \left(x - 10\right)$ and perform long division.

$\left(x - \alpha\right) \left(x - \beta\right) = - \frac{1000 - {x}^{3}}{x - 10}$

$= \frac{{x}^{3} - 1000}{x - 10}$

$= \frac{{x}^{3} \textcolor{g r e e n}{- 10 {x}^{2}}}{x - 10} + \frac{\textcolor{g r e e n}{10 {x}^{2}} - 1000}{x - 10}$

$= {x}^{2} + \frac{10 {x}^{2} \textcolor{b l u e}{- 100 x}}{x - 10} + \frac{\textcolor{b l u e}{100 x} - 1000}{x - 10}$

$= {x}^{2} + 10 x + 100$

Therefore, $1000 - {x}^{3} = - \left(x - 10\right) \left({x}^{2} + 10 x + 100\right)$

Note: It may be useful to memorize ${a}^{3} - {b}^{3} = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$