Jasonbradley
Home

(x^4+4x^3-x-4)/(x^3-1)

2 min read Jun 17, 2024
(x^4+4x^3-x-4)/(x^3-1)

Simplifying the Rational Expression (x^4+4x^3-x-4)/(x^3-1)

This article will guide you through the process of simplifying the rational expression (x^4+4x^3-x-4)/(x^3-1). We'll utilize techniques like polynomial long division and factoring to arrive at a simplified form.

Step 1: Polynomial Long Division

Since the degree of the numerator (4) is greater than the degree of the denominator (3), we can use polynomial long division to simplify the expression.

           x + 4 
     x^3 - 1 | x^4 + 4x^3  - x - 4
             -(x^4      - x) 
             ------------------
                  4x^3         - 4
                  -(4x^3 - 4)
                  --------------
                          0

Therefore, we find that:

(x^4+4x^3-x-4)/(x^3-1) = x + 4

Step 2: Factoring (Optional)

Although the expression is already simplified, we can further analyze the denominator by factoring it:

(x^3 - 1) = (x - 1)(x^2 + x + 1)

The quadratic factor (x^2 + x + 1) doesn't factor further over the real numbers.

Therefore, the complete factorization of the original expression would be:

(x^4+4x^3-x-4)/(x^3-1) = (x + 4)(x - 1)(x^2 + x + 1)

Conclusion

We've successfully simplified the rational expression (x^4+4x^3-x-4)/(x^3-1) using polynomial long division. The simplified form is x + 4. We also explored the factorization of the denominator for a more comprehensive understanding of the expression.

Featured Posts