Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$8y^3-125z^3$

Short answer

The factored form of the given expression is $(2y-5z)(4y^2+10yz+25z^2)$.

Step-by-step solution

Step 1. Given information.

The given expression is:
$8y^3-125z^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}8y^3-125z^3& =& {\left(2y\right)}^3-{\left(5z\right)}^3\\ & =& (2y-5z)\left({\left(2y\right)}^2+\left(2y\right)\left(5z\right)+{\left(5z\right)}^2\right)\left[\because a^3-b^3=(a-b)(a^2+ab+b^2)\right]\\ & =& (2y-5z)\left({4y}^2+10yz+25z^2\right)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(2y-5z)(4y^2+10yz+25z^2)$.