The expression to factor,
[tex]x^4-81[/tex]We can use the formula shown below to simplify the expression.
Addition Distributive Law
[tex]a^2-b^2=(a+b)(a-b)[/tex]In order to use this law to simplify, let's re-arrange our expression given,
[tex]\begin{gathered} x^4-81 \\ \text{This can be written as:} \\ (x^2)^2-(9)^2 \end{gathered}[/tex]This form of the expression is perfect to use the addition distributive law upon.
Using the rule, we can write the expression as,
[tex]\begin{gathered} (x^2)^2-(9)^2 \\ =(x^2+9)(x^2-9) \end{gathered}[/tex]This is not the fully factored form. Because we can use the same rule to further simplify the term (x^2 - 9).
Let's write it in the form:
[tex]\begin{gathered} (x^2-9) \\ \text{This can be written as:} \\ (x)^2-(3)^2 \end{gathered}[/tex]The image below clarifies this,
So, this can be written as:
[tex]\begin{gathered} x^4-81 \\ =(x^2)^2-(9)^2 \\ =(x^2+9)(x^2-9) \\ =(x^2+9)((x)^2-(3)^2) \\ =(x^2+9)(x+3)(x-3) \end{gathered}[/tex]This is the fully factored form.
