%2F(x%2B1).jpeg "(x^3+1)/(x+1)")
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.