91Ó°ÊÓ

Polynomial division is a method used for dividing a polynomial by another polynomial or a monomial. In this exercise, we are dividing a polynomial consisting of multiple terms structured with coefficients and variables. Each term is represented with exponents. Our main aim is to find a quotient easily.
In polynomial division, each component of the term—numerical coefficients, and variables with exponents—is simplified separately. This makes the process more manageable and keeps the computations clean. The given expression requires us to break down both the coefficients and variables.
We began by simplifying the coefficients using basic arithmetic division. Dividing a negative by a negative results in a positive number, which simplified our coefficient from \(-48\) to \(8\). This sets the stage for tackling variables.

Simplifying expressions involves breaking down complex algebraic expressions into simpler forms to make them more understandable and easier to work with. In this exercise, we aimed to simplify each part of the given algebraic fraction separately.
The initial expression \(\frac{-48 a^{3} b c^{5}}{-6 a^{2} c^{4}}\) contains both numeric and variable components that need simplification. For the numerical part, performing the division \(-48 \/ -6\) was our first step, giving us the simplified coefficient \(8\).
For the variables, we handled each letter individually, treating their powers through division rules concerning exponents. This simplifies the appearance of the variables and makes calculations and further operations more efficient, resulting in the final neat expression \(8abc\).

Variable exponents are used when variables in algebra are raised to a power, and they follow specific mathematical rules. In this context, our expression includes the variables \(a\), \(b\), and \(c\) raised to varying powers. Understanding how to work with them is crucial when simplifying expressions.
When dividing variables with exponents, the exponent of the divisor is subtracted from that of the dividend. For instance, in dividing \(a^3\) by \(a^2\), we subtract the exponents (3 - 2) to get \(a^1\) or simply \(a\). Similarly, \(c^5\) divided by \(c^4\) results in \(c^{1} = c\).
The variable \(b\) remains unchanged as it is present only in the numerator. Grasping these rules helps in efficiently reducing expressions with multiple variables and their exponents, aiding in creating simplified expressions like \(8abc\).