Question

In the following exercises, factor completely.

$32x^5{y}^2-162x{y}^2$

Short answer

On factoring, we get $2x{y}^2\left(4x^2+9\right)\left(2x+3\right)\left(2x-3\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$32x^5{y}^2-162x{y}^2$

Factor out the GCF,

$=2x{y}^2\left(16x^4-81\right)$

Write using the difference of square pattern,

$$\begin{gathered} =2x{y}^2\{{\left(4x^2\right)}^2-9^2\} \\ =2x{y}^2\left(4x^2+9\right)\left(4x^2-9\right) \end{gathered}$$

Write using the sum of square pattern,

$$\begin{gathered} =2x{y}^2\left(4x^2+9\right)\left\{{\left(2x\right)}^2-3^2\right\} \\ =2x{y}^2\left(4x^2+9\right)\left(2x+3\right)\left(2x-3\right) \end{gathered}$$

Step 2. Check the answers.

Check by multiplying,

$$\begin{gathered} =2x{y}^2\left(4x^2+9\right)\left(2x+3\right)\left(2x-3\right) \\ =2x{y}^2\left(4x^2+9\right)\left(4x^2-6x+6x-9\right) \\ =2x{y}^2\left(4x^2+9\right)\left(4x^2-9\right) \\ =2x{y}^2\left(16x^4-36x^2+36x^2-81\right) \\ =2x{y}^2\left(16x^4-81\right) \\ =32x^5{y}^2-162x{y}^2 \end{gathered}$$

This is true.