(x^3+1)/(x+1)

2 min read Jun 17, 2024
(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.

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