To factor the expression \( 25x^4 - 1 \), you can recognize that it is a difference of squares.
The difference of squares can be factored using the identity:
\[ a^2 - b^2 = (a - b)(a + b) \]
In this case, we can rewrite \( 25x^4 - 1 \) as:
\[ (5x^2)^2 - 1^2 \]
Now, apply the difference of squares formula:
\[ (5x^2 - 1)(5x^2 + 1) \]
Now, observe that \( 5x^2 - 1 \) itself can be factored further, since it is also a difference of squares:
\[ 5x^2 - 1 = (5x^2 - 1^2) \]
So we can write:
\[ (5x - 1)(5x + 1) \]
Thus, the complete factorization of \( 25x^4 - 1 \) is:
\[ (5x - 1)(5x + 1)(5x^2 + 1) \]
The final factored form of \( 25x^4 - 1 \) is:
\[ (5x - 1)(5x + 1)(5x^2 + 1) \]
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