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To factor the binomial \( 25x^2 - 1 \) completely, let's identify if it can be factored as a difference of squares.

The expression \( 25x^2 - 1 \) is indeed a difference of squares because it can be written as:
\[
(5x)^2 - 1^2
\]

According to the difference of squares formula:
\[
a^2 - b^2 = (a + b)(a - b)
\]
we can factor \( 25x^2 - 1 \) as:
\[
(5x + 1)(5x - 1)
\]

So, the fully factored form of \( 25x^2 - 1 \) is:
\[
(5x + 1)(5x - 1)
\]

Would you like a further breakdown of each step?

### Related Questions
1. How would you factor \( 16x^2 - 9 \) using the difference of squares?
2. Can a sum of squares, such as \( x^2 + 4 \), be factored in real numbers?
3. What are the key characteristics of the difference of squares?
4. How would you solve \( 25x^2 - 1 = 0 \) after factoring?
5. Can a difference of cubes, like \( x^3 - 8 \), be factored similarly?

### Tip
Always look for a common factor before applying the difference of squares technique.

Factor the following binomial completely: 25x^2 - 1

Learn how to factor the binomial 25x^2 - 1 completely by applying the difference of squares technique. This solution is ideal for students in Grades 8-10 who want to understand algebraic factoring methods.

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