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In algebra, we work with expressions that contain variables, numbers, and arithmetic operations. An algebraic expression is a mathematical phrase that can involve adding, subtracting, multiplying, and dividing.
Algebraic expressions can range from simple, like \(3x + 2\), to complex ones, like \(4x^2 - 7x + 5\). In the given exercise, \(3x - 12\) and \(15x\) are algebraic expressions. The goal is to simplify these expressions to their simplest form.
To simplify, it often helps to combine like terms and perform operations indicated by the expression. This process makes it easier to solve equations or other mathematical problems.
Overall, understanding how to manipulate algebraic expressions is a key skill in algebra.

Factoring involves breaking down an algebraic expression into simpler pieces that can be multiplied together to give the original expression. This is particularly useful for simplifying fractions.
In our example, the numerator \(3x - 12\) can be factored by looking for a common factor. Both terms \(3x\) and \(-12\) have a common factor of 3. We can then write the numerator as \(3(x - 4)\).
Factoring makes simplifying fractions easier because it helps to reveal common factors between the numerator and the denominator. By removing these common factors, we reduce the expression to its simplest form.
Mastering factoring is vital since it is widely used in solving quadratic equations, integrating functions, and simplifying algebraic fractions.

Understanding the roles of the numerator and denominator is essential when working with fractions. The numerator is the top part of a fraction, and the denominator is the bottom part.
In the expression \(\frac{3x - 12}{15x}\), \(3x - 12\) is the numerator, and \(15x\) is the denominator. Simplifying algebraic fractions often involves factoring both the numerator and the denominator to find and remove common factors.
In our solution, after factoring the numerator, we rewrite the expression as \(\frac{3(x - 4)}{15x}\). By identifying the common factor of 3 in both the numerator and the denominator, we can cancel out this common factor:
\(\frac{3(x-4)}{15x} = \frac{x-4}{5x}\).
This cancellation simplifies the fraction, making it more manageable and easier to work with in further calculations.