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When subtracting fractions, having a common denominator is essential. The denominator is the bottom part of a fraction that names the total parts in a whole. Two fractions must share the same denominator before performing the operation of addition or subtraction. In the exercise, both fractions already have a common denominator of \( p \), meaning we can proceed directly to subtracting the numerators. This is often the case with like fractions, which share the same denominator from the beginning.

Finding a common denominator can often include steps like identifying the least common multiple (LCM) of two numbers. However, since our fractions are already aligned with \( p \), this process is simplified, allowing us to skip several unnecessary steps. Therefore, a common denominator simplifies the arithmetic process by aligning the pieces of one whole, making processes easier to handle.

Numerator subtraction is the process wherein you subtract the top numbers of two fractions after ensuring they share a common denominator. In the current example, we subtract the numerators directly as our denominators are already identical:

\[ \frac{3}{p} - \frac{9}{p} = \frac{3 - 9}{p} \]

This simply means we take the value 3 and subtract it by 9, getting \(-6\) in total. This value becomes the new numerator of the resulting fraction. It's important to always subtract the numerators while keeping the denominator the same, as it represents the division of the parts we are working with. As straightforward as this might sound, taking care when subtracting negative numbers can help avoid mistakes.

Simplifying fractions is the last step which involves reducing the fraction to its simplest form. In this example, our result \( \frac{-6}{p} \) doesn't require further simplification beyond performing the subtraction in the numerator:\[ \frac{3 - 9}{p} = \frac{-6}{p} \].

To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). In cases where the fraction's numerator and denominator share a common divisible factor, implementing this can reduce the fraction to its smallest terms. However, with variables like \( p \), simplification primarily pertains to ensuring arithmetic operations in the numerator are accurate. Therefore, ensure all calculations within fractions are done thoroughly for correct results.