Question
Factor completely.
25, x, to the power 4 , minus, 1
25x
4
−1
25, x, to the power 4 , minus, 1
25x
4
−1
Answers
GPT-4o mini
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:
\[
(5x - 1)(5x + 1)(5x^2 + 1)
\]
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:
\[
(5x - 1)(5x + 1)(5x^2 + 1)
\]