## Simplifying the Expression (x^3 + 1) / (x + 1)

The expression (x^3 + 1) / (x + 1) can be simplified using various methods. Here are two common approaches:

### 1. Factoring and Cancellation

**Recognizing the pattern:**The numerator, x^3 + 1, is a sum of cubes. This can be factored as (x + 1)(x^2 - x + 1).**Factoring the expression:**

(x^3 + 1) / (x + 1) = [(x + 1)(x^2 - x + 1)] / (x + 1)**Canceling common factors:**Since (x + 1) appears in both the numerator and denominator, we can cancel them out.**Simplified form:**The simplified expression is**x^2 - x + 1**.

This simplification is valid for all values of x except for x = -1, where the original expression is undefined.

### 2. Polynomial Long Division

**Setting up the division:**We can perform polynomial long division with (x + 1) as the divisor and (x^3 + 1) as the dividend.**Performing the division:**The process involves finding the quotient and remainder. In this case, the quotient is x^2 - x + 1 and the remainder is 0.**Result:**The result of the division is**x^2 - x + 1**.

Both methods lead to the same simplified form: **x^2 - x + 1**.

**Note:** While both methods work, factoring and cancellation is generally considered a more efficient approach for this particular expression.