Q. 251 - Intermediate Algebra

Question

In the following exercises, factor completely.   

$5x^5{y}^2-80x{y}^2$

Step-by-Step Solution

Answer

The required factors are  $5x{y}^2(x^2+4)(x-2)(x+2)$

Step 1. Given Information

The given expression is $5x^5{y}^2-80x{y}^2$

Step 2. Calculation

We will find the factors by using difference of square patterns,

$$\begin{gathered} 5x^5{y}^2-80x{y}^2=5x{y}^2(x^4-16) \\ =5x{y}^2({\left(x^2\right)}^2-4^2) \\ =5x{y}^2\left[\right(x^2-4\left)\right(x^2+4\left)\right] \\ =5x{y}^2(x^2+4)(x^2-2^2) \\ =5x{y}^2(x^2+4)(x-2)(x+2) \end{gathered}$$

Step 3. Check

We will verify the factors by multiplying the factors.

$$\begin{gathered} 5x{y}^2(x^2+4)(x-2)(x+2)=5x{y}^2(x^2+4)(x^2-2^2) \\ =5x{y}^2\left[\right(x^2-4\left)\right(x^2+4\left)\right] \\ =5x{y}^2({\left(x^2\right)}^2-4^2) \\ =5x{y}^2(x^4-16) \\ =5x^5{y}^2-80x{y}^2 \end{gathered}$$