(x^3-1)/(x-1)

2 min read Jun 17, 2024
(x^3-1)/(x-1)

Factoring and Simplifying (x^3 - 1) / (x - 1)

The expression (x^3 - 1) / (x - 1) represents a rational function, where the numerator and denominator are both polynomials. We can simplify this expression by factoring and canceling common factors.

Factoring the Numerator

The numerator (x^3 - 1) is a difference of cubes. We can factor it using the following formula:

a³ - b³ = (a - b)(a² + ab + b²)

In our case, a = x and b = 1. Applying the formula, we get:

x³ - 1 = (x - 1)(x² + x + 1)

Simplifying the Expression

Now, let's substitute the factored numerator back into the original expression:

(x³ - 1) / (x - 1) = [(x - 1)(x² + x + 1)] / (x - 1)

Since (x - 1) appears in both the numerator and denominator, we can cancel them out (assuming x ≠ 1):

[(x - 1)(x² + x + 1)] / (x - 1) = x² + x + 1

Conclusion

Therefore, the simplified form of (x³ - 1) / (x - 1) is x² + x + 1, provided that x ≠ 1.

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