Question

Instructions: Factor the following expressions completely, show your work and explain each step when factoring each expression.1) x^4 - 812) p^2 - 4p = 103) 3x^2 - 2x = -12

Answer 1

1)

The expression to factor,

`x^4-81`

We can use the formula shown below to simplify the expression.

Addition Distributive Law

`a^2-b^2=(a+b)(a-b)`

In order to use this law to simplify, let's re-arrange our expression given,

`\begin{gathered} x^4-81 \\ \text{This can be written as:} \\ (x^2)^2-(9)^2 \end{gathered}`

This form of the expression is perfect to use the addition distributive law upon.

Using the rule, we can write the expression as,

`\begin{gathered} (x^2)^2-(9)^2 \\ =(x^2+9)(x^2-9) \end{gathered}`

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:

`\begin{gathered} (x^2-9) \\ \text{This can be written as:} \\ (x)^2-(3)^2 \end{gathered}`

The image below clarifies this,

So, this can be written as:

`\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}`

This is the fully factored form.