Certainly! Let's factorize the given polynomial step by step.
The polynomial given is:
[tex]\[ 9x^2 - 16 \][/tex]
We need to factorize it. This polynomial is a difference of squares. The difference of squares formula is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In this polynomial, we can identify [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] as follows:
[tex]\[ 9x^2 = (3x)^2 \][/tex]
[tex]\[ 16 = 4^2 \][/tex]
So, we can write the polynomial as:
[tex]\[ 9x^2 - 16 = (3x)^2 - 4^2 \][/tex]
By applying the difference of squares formula, we get:
[tex]\[ (3x)^2 - 4^2 = (3x - 4)(3x + 4) \][/tex]
Hence, the factorization of the polynomial [tex]\( 9x^2 - 16 \)[/tex] is:
[tex]\[ (3x - 4)(3x + 4) \][/tex]
Now, let's match this result with the given options:
A. [tex]\((9x - 4)(x - 4)\)[/tex]
B. [tex]\((3x - 4)(3x - 4)\)[/tex]
C. [tex]\((9x + 4)(x - 4)\)[/tex]
D. [tex]\((3x + 4)(3x - 4)\)[/tex]
Among these options, the correct match is option D:
[tex]\[ (3x + 4)(3x - 4) \][/tex]
Therefore, the factorization is:
[tex]\[ \boxed{(3x + 4)(3x - 4)} \][/tex]
Hence, the correct answer is option D.
