Q. 208 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$-2x^3{y}^2-16y^5$

Step-by-Step Solution

Answer

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

Step 1. Given information.

The given expression is:
$-2x^3{y}^2-16y^5$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}-2x^3{y}^2-16y^5& =& -2y^2(x^3+8y^3)\\ & =& -2y^2(x^3+{\left(2y\right)}^3) \\ & =& -2y^2(x+2y)(x^2-(x\left)\right(2y)+{\left(2y\right)}^2) \left[\because a^3+b^3=(a+b)(a^2-ab+b^2)\right]\\ & =& -2y^2(x+2y)(x^2-2xy+4y^2)\end{array}$

Step 3. Conclusion.

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