Simplifying the Rational Expression (6x^3  x^2 + 12x) / (x^2 + 2)
This article explores the simplification of the rational expression (6x^3  x^2 + 12x) / (x^2 + 2). We will break down the process stepbystep and examine the resulting simplified form.
Understanding Rational Expressions
A rational expression is a fraction where both the numerator and denominator are polynomials. In our case, the numerator is 6x^3  x^2 + 12x and the denominator is x^2 + 2.
Simplifying the Expression
The key to simplifying rational expressions is to factor both the numerator and denominator as much as possible and then cancel out any common factors.

Factoring the Numerator:
 We can factor out a common factor of x from the numerator: (6x^3  x^2 + 12x) = x(6x^2  x + 12)

Factoring the Denominator:
 The denominator (x^2 + 2) cannot be factored further using real numbers.

Cancellation:
 We cannot cancel out any factors between the numerator and denominator because there are no common factors.
Resulting Simplified Form
Therefore, the simplified form of the rational expression (6x^3  x^2 + 12x) / (x^2 + 2) is (x(6x^2  x + 12)) / (x^2 + 2).
Conclusion
While we were unable to cancel out any factors in this specific case, the process of factoring and canceling common factors is a fundamental technique for simplifying rational expressions. Understanding this process allows us to simplify complex expressions and work with them more efficiently.