%2F(x%5E2%2B2).jpeg "(6x^3-x^2+12x)/(x^2+2)")
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 step-by-step 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.
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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)
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Factoring the Denominator:
- The denominator (x^2 + 2) cannot be factored further using real numbers.
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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.