Sure, let's go through the process of factoring the expression [tex]\(81x^4 - 16\)[/tex] step-by-step.
1. Identify the type of expression: The given expression [tex]\(81x^4 - 16\)[/tex] is a difference of squares. Recall that [tex]\(a^2 - b^2\)[/tex] factors into [tex]\((a - b)(a + b)\)[/tex].
2. Rewrite the expression as a difference of squares:
[tex]\[
81x^4 - 16 = (9x^2)^2 - 4^2
\][/tex]
3. Apply the difference of squares formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(a = 9x^2\)[/tex] and [tex]\(b = 4\)[/tex]. Therefore, we can rewrite the expression as:
[tex]\[
(9x^2 - 4)(9x^2 + 4)
\][/tex]
4. Factor each term if possible: Now, focus on the term [tex]\(9x^2 - 4\)[/tex] which is another difference of squares. Specifically:
[tex]\[
9x^2 - 4 = (3x)^2 - 2^2
\][/tex]
Again, using the difference of squares:
[tex]\[
(3x)^2 - 2^2 = (3x - 2)(3x + 2)
\][/tex]
5. Combine all factors: Now that we have factored [tex]\(9x^2 - 4\)[/tex], we can write the complete factorization of the original expression:
[tex]\[
81x^4 - 16 = (3x - 2)(3x + 2)(9x^2 + 4)
\][/tex]
Thus, the completely factored form of [tex]\(81x^4 - 16\)[/tex] is [tex]\((3x - 2)(3x + 2)(9x^2 + 4)\)[/tex]. Therefore, the correct answer is:
[tex]\[
(3x - 2)(3x + 2)(9x^2 + 4)
\][/tex]
