To factor the polynomial [tex]\(x^2 - 49\)[/tex] completely, we need to recognize that it is a difference of squares.
A difference of squares is a quadratic expression of the form [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a - b)(a + b)\)[/tex]. For the given polynomial:
[tex]\[
x^2 - 49
\][/tex]
We can see that:
[tex]\[
a^2 = x^2 \quad \text{and} \quad b^2 = 49
\][/tex]
Since [tex]\(49\)[/tex] is a perfect square, we can write it as:
[tex]\[
49 = 7^2
\][/tex]
Now, we apply the difference of squares formula:
[tex]\[
x^2 - 49 = x^2 - 7^2
\][/tex]
Using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], we substitute [tex]\(a = x\)[/tex] and [tex]\(b = 7\)[/tex]:
[tex]\[
x^2 - 7^2 = (x - 7)(x + 7)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(x^2 - 49\)[/tex] is:
[tex]\[
(x - 7)(x + 7)
\][/tex]
Looking at the given options:
A. [tex]\((x - 7)(x - 7)\)[/tex]
B. [tex]\((x + 7)(x - 7)\)[/tex]
C. [tex]\((x - 49)(x - 49)\)[/tex]
D. [tex]\((x + 49)(x - 49)\)[/tex]
The correct answer is:
B. [tex]\((x + 7)(x - 7)\)[/tex]
