Question

Factor:$1000m^3-125n^3$.

Short answer

Factors of$1000m^3-125n^3$are$\left(10m-5n\right)\left(100m^2+50mn+25n^2\right)$.

Step-by-step solution

Step 1. Given information.

Given an expression$1000m^3-125n^3$.

Step 2. Check if the binomial is a difference and write the terms as cubes.

Consider $a=10m$and $b=5n$.

It can be seen that $1000m^2-125n^2={\left(10m\right)}^2-{\left(5n\right)}^2$.

Therefore, $1000m^3-125n^3$is a difference of cubes.

Step 3. Use the difference of cubes pattern and simplify.

Note that $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$..

Factor the expression as follows.

$\begin{array}{rcl}1000m^3-125n^2& =& {\left(10m\right)}^3-{\left(5n\right)}^2\\ & =& \left(10m-5n\right)\left({\left(10m\right)}^2+50mn+{\left(5n\right)}^2\right)\\ & =& \left(10m-5n\right)\left(100m^2+50mn+25n^2\right)\end{array}$

Step 4. Check by multiplying the factors.

Verify as follows.

$\begin{array}{rcl}\left(10m-5n\right)\left(100m^2+50mn+25n^2\right)& =& 10m\left(100m^2+50mn+25n^2\right)-5n\left(100m^2+50mn+25n^2\right)\\ & =& 1000m^3+500m^{2}n+250n^{2}m-500n{m}^2-250m{n}^2-125n^3\\ & =& 1000m^3-125n^3\end{array}$