Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$2x^2-16x^2{y}^3$

Short answer

The factored form of the given expression is$2x^2(2y-1)(4y^2+2y+1)$.

Step-by-step solution

Step 1. Given information.

The given expression is:
$2x^2-16x^2{y}^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}2x^2-16x^2{y}^3& =& 2x^2(1-8y^3)\\ & =& 2x^2(1^3-{\left(2y\right)}^3)\\ & =& 2x^2(1-2y)(1^2+(1\left)\right(2y)+{\left(2y\right)}^2)\left[\because a^3-b^3=(a-b)(a^2+ab+b^2)\right]\\ & =& 2x^2(1-2y)(1+2y+4y^2)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is$2x^2(2y-1)(4y^2+2y+1)$.