%2F(3a-2).jpeg "(6a^2-3a+9)/(3a-2)")
Simplifying Rational Expressions: (6a^2 - 3a + 9) / (3a - 2)
This article will guide you through the process of simplifying the rational expression (6a^2 - 3a + 9) / (3a - 2).
Understanding Rational Expressions
A rational expression is a fraction where the numerator and denominator are polynomials. To simplify these expressions, we aim to factor both the numerator and denominator and then cancel out any common factors.
Factoring the Numerator
The numerator, 6a^2 - 3a + 9, doesn't factor easily using traditional methods. It doesn't fit the pattern of a difference of squares or a perfect square trinomial. However, we can factor out a common factor of 3:
3(2a^2 - a + 3)
Factoring the Denominator
The denominator, 3a - 2, is already in its simplest factored form.
Simplifying the Expression
Now we can rewrite the original expression with its factored components:
(3(2a^2 - a + 3)) / (3a - 2)
Unfortunately, there are no common factors between the numerator and denominator. This means the expression is already in its simplest form.
Conclusion
While we couldn't simplify the expression further, we successfully factored the numerator and identified that no common factors exist between the numerator and denominator. This process highlights the importance of factoring in simplifying rational expressions and recognizing when an expression is already in its simplest form.