Q. 250 - Intermediate Algebra

Question

In the following exercises, factor completely.   

$m^4-81$

Step-by-Step Solution

Answer

The required factors are $(m^2+9)(m-3)(m+3)$

Step 1. Given Information

The given expression is $m^4-81$

Step 2. Calculation

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

$$\begin{gathered} m^4-81={\left(m^2\right)}^2-9^2 \\ =(m^2+9)(m^2-9) \\ =(m^2+9)(m^2-3^2) \\ =(m^2+9)(m-3)(m+3) \end{gathered}$$

Step 3. Check

We will verify the factors by multiplying the factors.

$$\begin{gathered} (m^2+9)(m-3)(m+3)=(m^2+9)(m^2-3^2) \\ =(m^2+9)(m^2-9) \\ ={\left(m^2\right)}^2-9^2 \\ =m^4-81 \end{gathered}$$