To factor the polynomial [tex]\(81x^2 - 16\)[/tex] completely, follow these steps:
1. Recognize the Form:
Notice that [tex]\(81x^2 - 16\)[/tex] corresponds to the general form of a difference of squares, which is [tex]\(a^2 - b^2\)[/tex].
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Here, [tex]\(81x^2\)[/tex] is the square of [tex]\(9x\)[/tex] and [tex]\(16\)[/tex] is the square of [tex]\(4\)[/tex]. So, we can set:
[tex]\[
a = 9x \quad \text{and} \quad b = 4
\][/tex]
3. Apply the Difference of Squares Formula:
The difference of squares formula states:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
Using the identified values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
81x^2 - 16 = (9x)^2 - 4^2 = (9x + 4)(9x - 4)
\][/tex]
Therefore, the complete factorization of [tex]\(81x^2 - 16\)[/tex] is:
[tex]\[
(9x + 4)(9x - 4)
\][/tex]
So, the correct answer is:
D. [tex]\((9x + 4)(9x - 4)\)[/tex]
