$\frac{16 \cdot x^2+21 x+54}{(x-y x)}$
The expression provided is \(\frac{16x^2 + 21x + 54}{x - yx}\). To understand its structure and behavior, we first examine the denominator, \(x - yx\). This expression can be factored by taking out the common factor \(x\), resulting in \(x(1 - y)\). Thus, the full expression is \(\frac{16x^2 + 21x + 54}{x(1 - y)}\). For this expression to be defined, the denominator must be non-zero, which requires \(x eq 0\) and \(y eq 1\).
To simplify the expression, we can distribute the \(x\) in the denominator to each term in the numerator: \[\frac{16x^2 + 21x + 54}{x(1 - y)} = \frac{1}{\left(1 - y\right)} \cdot \left(\frac{16x^2}{x} + \frac{21x}{x} + \frac{54}{x}\right).\] Simplifying the terms inside the parentheses gives \(\left(16x + 21 + \frac{54}{x}\right)\). Therefore, the expression can be written as \(\frac{16x + 21 + \frac{54}{x}}{1 - y}\). This form clearly shows how the expression behaves as \(x\) becomes very large or very small, as well as its dependence on the variable \(y\).
Consider a scenario where you are calculating the average cost per unit produced, where the total cost is represented by a quadratic function \(16x^2 + 21x + 54\) and the number of units is given by \(x - yx\). This algebraic form helps managers analyze how the unit cost fluctuates based on production volume (\(x\)) and a scaling efficiency factor (\(y\)). By simplifying the expression, the manager can quickly determine how changing the production scale or the efficiency factor will impact the final unit cost.