Let's factor the given polynomial [tex]\(4a^2 - 49\)[/tex].
Step-by-Step Solution:
1. Identify the form of the polynomial:
The given polynomial [tex]\(4a^2 - 49\)[/tex] is a difference of squares. A difference of squares is a term that can be written in the form [tex]\(A^2 - B^2\)[/tex].
2. Express the terms as squares:
We can write [tex]\(4a^2\)[/tex] as [tex]\((2a)^2\)[/tex] and [tex]\(49\)[/tex] as [tex]\(7^2\)[/tex]. Thus, the polynomial [tex]\(4a^2 - 49\)[/tex] can be rewritten as:
[tex]\[
(2a)^2 - 7^2
\][/tex]
3. Apply the difference of squares formula:
The difference of squares formula states [tex]\(A^2 - B^2 = (A - B)(A + B)\)[/tex]. Here, [tex]\(A = 2a\)[/tex] and [tex]\(B = 7\)[/tex]. Using the formula, we get:
[tex]\[
(2a)^2 - 7^2 = (2a - 7)(2a + 7)
\][/tex]
4. Write the final factored form:
Therefore, the factored form of the polynomial [tex]\(4a^2 - 49\)[/tex] is:
[tex]\[
(2a - 7)(2a + 7)
\][/tex]
Thus, the correct factorization for the polynomial [tex]\(4a^2 - 49\)[/tex] is: [tex]\((2a - 7)(2a + 7)\)[/tex].
