## Understanding the Expression (a-1)(a+1)(a^2+1)

The expression (a-1)(a+1)(a^2+1) is a fascinating algebraic pattern that can be simplified and explored in various ways. Let's delve into its properties and applications.

### Recognizing the Pattern

The expression appears to be a product of four factors:

**(a-1) and (a+1):**These are two binomials representing the difference of squares pattern.**(a^2+1):**This represents the sum of squares.

### Simplifying the Expression

We can simplify the expression by applying the difference of squares pattern:

**(a-1)(a+1) = a^2 - 1**

Substituting this back into the original expression, we get:

**(a^2 - 1)(a^2 + 1)**

Now, applying the difference of squares pattern again:

**(a^2 - 1)(a^2 + 1) = a^4 - 1**

Therefore, the simplified form of the expression (a-1)(a+1)(a^2+1) is **a^4 - 1**.

### Applications and Significance

This expression has several applications in different areas of mathematics:

**Factoring polynomials:**It provides a useful pattern for factoring polynomials of the form a^4 - 1.**Solving equations:**It can be used to solve equations involving expressions like a^4 - 1.**Calculus:**It can be used to calculate derivatives and integrals involving expressions like a^4 - 1.

### Example

Let's consider an example where a = 2.

**(a-1)(a+1)(a^2+1) = (2-1)(2+1)(2^2+1)****= (1)(3)(5)****= 15**

We can also verify this result using the simplified form:

**a^4 - 1 = 2^4 - 1 = 16 - 1 = 15**

Therefore, the value of the expression for a = 2 is 15, which is consistent with both the original and simplified forms.

### Conclusion

The expression (a-1)(a+1)(a^2+1) is an interesting algebraic pattern that can be simplified and applied in various mathematical contexts. Its understanding helps in factoring polynomials, solving equations, and performing other calculations. By recognizing the difference of squares pattern and applying it appropriately, we can effectively simplify and analyze this expression.