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To simplify a rational expression, the first step is often to factor both the numerator and the denominator. Factoring is the process of breaking down a polynomial into simpler terms (factors) that, when multiplied together, give you the original polynomial. For example, consider the polynomial in the numerator of the given exercise: \(3z^2 + z\). We look for the greatest common factor, which is the largest term that can evenly divide each term of the polynomial. Here, the common factor is \(z\), and when we factor \(z\) out, we get \(z(3z + 1)\). This step simplifies the expression significantly by breaking it into components that are easier to work with.

Once both the numerator and the denominator are factored, the next step is to simplify the rational expression by canceling common factors. Common factors are terms that appear in both the numerator and the denominator. In our exercise, after factoring, we obtained: \(\frac{z(3z + 1)}{6(3z + 1)}\). The term \(3z + 1\) is common to both the numerator and the denominator. By canceling out these common factors, we are left with: \(\frac{z}{6}\). It's crucial to remember that only terms that are multiplied (not added or subtracted) can be canceled this way. This step effectively reduces the expression to its simplest form.

Understanding the roles of the numerator and the denominator in a rational expression is key to simplifying it. The numerator is the top part of the fraction, and the denominator is the bottom part. In the given problem, the numerator is \(3z^2 + z\) and the denominator is \(18z + 6\). Simplifying each part separately can make the overall process less daunting. For the numerator: \(3z^2 + z = z(3z + 1)\). For the denominator: \(18z + 6 = 6(3z + 1)\). By factoring both parts, we prepare the expression for canceling any common factors, thus streamlining the original rational expression into its simplest form \(\frac{z}{6}\). Comprehending how to handle the numerator and denominator independently is a foundational skill in algebra.