If $3 x$ is one factor of $3 x^2-9 x$, what is the other factor?
To find the other factor of the expression \(3x^2 - 9x\), we are given that \(3x\) is one of the factors. This means there exists some factor, let's call it \(y\), such that the product of \(3x\) and \(y\) is equal to the original expression. We can represent this relationship with the equation:
\[3x \cdot y = 3x^2 - 9x\]
To find the value of \(y\), we need to isolate it in the equation. We do this by dividing both sides of the equation by \(3x\):
\[y = \frac{3x^2 - 9x}{3x}\]
We can simplify this fraction by dividing each term in the numerator by the denominator:
\[y = \frac{3x^2}{3x} - \frac{9x}{3x}\]
Calculating each term separately:
\[\frac{3x^2}{3x} = x\]
\[\frac{9x}{3x} = 3\]
Substituting these back, we get:
\[y = x - 3\]
Comparing our result \(y = x - 3\) with the given options:
The result \(x - 3\) matches Option D. Therefore, if \(3x\) is one factor, the other factor is \(x - 3\). Thus, the correct option is \(\boxed{\text{D}}\).
Factoring is a fundamental skill in algebra used to simplify expressions and solve equations. For example, if an engineer is calculating the area of a rectangular plot of land represented by the expression \(3x^2 - 9x\) and knows that the width is \(3x\), they can use factoring to determine the length. In this case, the length would be \(x - 3\). This helps in understanding dimensions in design and construction tasks, making calculations much more manageable.