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Factor the following binomial completely:

[tex]x^2 - 169y^2[/tex]

Select the correct choice below and, if necessary, fill in the blank:

A. [tex]x^2 - 169y^2 = \square[/tex] (Simplify your answer.)
B. [tex]x^2 - 169y^2[/tex] is prime.

Answer :

We start with the expression

[tex]$$
x^2 - 169y^2.
$$[/tex]

Notice that [tex]$169$[/tex] is a perfect square since

[tex]$$
169 = 13^2.
$$[/tex]

Thus, we can rewrite the expression as

[tex]$$
x^2 - (13y)^2.
$$[/tex]

This expression is now in the form of a difference of two squares, which is given by the formula

[tex]$$
a^2 - b^2 = (a - b)(a + b).
$$[/tex]

Here, we take

[tex]$$
a = x \quad \text{and} \quad b = 13y.
$$[/tex]

Applying the formula, we have

[tex]$$
x^2 - (13y)^2 = (x - 13y)(x + 13y).
$$[/tex]

Thus, the factorization of [tex]$x^2 - 169y^2$[/tex] is

[tex]$$
\boxed{(x-13y)(x+13y)}.
$$[/tex]

This is the completely factored form of the given binomial.

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