To simplify the expression [tex]\((a+1)(a-1)\left(a^2+1\right)\left(a^2+1\right)\)[/tex], let us follow a systematic, step-by-step approach.
1. Identify Standard Algebraic Identities:
Notice that we have [tex]\((a + 1)(a - 1)\)[/tex]. This is a standard difference of squares, which can be simplified as:
[tex]\[
(a + 1)(a - 1) = a^2 - 1
\][/tex]
2. Rewrite the Expression:
Given the algebraic identity, we can rewrite the initial expression:
[tex]\[
(a + 1)(a - 1)(a^2 + 1)(a^2 + 1) = (a^2 - 1)(a^2 + 1)^2
\][/tex]
3. Expand the Square:
Next, let's expand [tex]\((a^2 + 1)^2\)[/tex]. This is a square of a binomial, which follows the pattern [tex]\((x + y)^2 = x^2 + 2xy + y^2\)[/tex]. Applying this pattern, we get:
[tex]\[
(a^2 + 1)^2 = (a^2)^2 + 2(a^2)(1) + 1^2 = a^4 + 2a^2 + 1
\][/tex]
4. Substitute Back:
Now, substitute [tex]\(a^4 + 2a^2 + 1\)[/tex] back into the simplified expression:
[tex]\[
(a^2 - 1)(a^4 + 2a^2 + 1)
\][/tex]
5. Distribute and Multiply:
Now, distribute [tex]\(a^2 - 1\)[/tex] across each term in the trinomial [tex]\(a^4 + 2a^2 + 1\)[/tex]:
[tex]\[
(a^2 - 1)(a^4 + 2a^2 + 1) = (a^2 - 1) \cdot a^4 + (a^2 - 1) \cdot 2a^2 + (a^2 - 1) \cdot 1
\][/tex]
6. Simplify Each Term:
[tex]\[
= a^6 - a^4 + 2a^4 - 2a^2 + a^2 - 1
\][/tex]
7. Combine Like Terms:
To combine like terms, we add or subtract the coefficients of the terms with the same powers of [tex]\(a\)[/tex]:
[tex]\[
a^6 + (2a^4 - a^4) + (-2a^2 + a^2) - 1 = a^6 + a^4 - a^2 - 1
\][/tex]
Thus, the simplified form of the expression [tex]\((a + 1)(a - 1)(a^2 + 1)(a^2 + 1)\)[/tex] is:
[tex]\[
a^6 + a^4 - a^2 - 1
\][/tex]
