Sure, let's factorize the expression [tex]\( a^4 - 1 \)[/tex] step-by-step.
1. Recognize the Difference of Squares:
[tex]\[
a^4 - 1 = (a^2)^2 - 1^2
\][/tex]
This expression is a difference of squares, which can be written as:
[tex]\[
(a^2 - 1)(a^2 + 1)
\][/tex]
2. Factorize [tex]\( a^2 - 1 \)[/tex]:
Notice that [tex]\( a^2 - 1 \)[/tex] itself is also a difference of squares:
[tex]\[
a^2 - 1 = (a - 1)(a + 1)
\][/tex]
3. Combine All Factors:
Substituting the factorized form of [tex]\( a^2 - 1 \)[/tex] back into the original expression gives:
[tex]\[
a^4 - 1 = (a - 1)(a + 1)(a^2 + 1)
\][/tex]
Thus, the fully factorized form of [tex]\( a^4 - 1 \)[/tex] is:
[tex]\[
(a - 1)(a + 1)(a^2 + 1)
\][/tex]
