To factor the polynomial expression [tex]\(5x^4 - 80\)[/tex] completely, follow these steps:
1. Identify the common factor:
First, observe that the terms [tex]\(5x^4\)[/tex] and [tex]\(80\)[/tex] share a common factor of 5. Factor out this common factor:
[tex]\[
5x^4 - 80 = 5(x^4 - 16)
\][/tex]
2. Factor the quadratic expression:
Next, we need to factor the expression inside the parentheses, [tex]\(x^4 - 16\)[/tex]. Notice that [tex]\(x^4 - 16\)[/tex] is a difference of squares, which can be written as:
[tex]\[
x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4)
\][/tex]
3. Factor the difference of squares further:
The expression [tex]\(x^2 - 4\)[/tex] is also a difference of squares. It can be factored as follows:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
So, substituting back:
[tex]\[
x^4 - 16 = (x - 2)(x + 2)(x^2 + 4)
\][/tex]
Now substitute this back into the expression we factored out initially:
[tex]\[
5(x^4 - 16) = 5((x - 2)(x + 2)(x^2 + 4))
\][/tex]
Thus, the completely factored form of [tex]\(5x^4 - 80\)[/tex] is:
[tex]\[
5(x - 2)(x + 2)(x^2 + 4)
\][/tex]
So, the correct answer is:
[tex]\[
5(x - 2)(x + 2)(x^2 + 4)
\][/tex]
