The greatest common factor of $3 m^2 n+12 m n^2$ is:
To find the greatest common factor (GCF) of the expression \(3m^2n + 12mn^2\), we examine the two terms individually: \(3m^2n\) and \(12mn^2\). The GCF is the largest monomial that can evenly divide both terms. We break this down by looking at the coefficients and the variables separately.
First, look at the coefficients of the two terms, which are 3 and 12. The factors of 3 are 1 and 3. The factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor of 3 and 12 is 3.
Next, examine the powers of the variable \(m\). The first term contains \(m^2\) and the second term contains \(m^1\). The greatest common factor for the power of a variable is the lowest power present in the terms, which is \(m^1\), or simply \(m\). Similarly, for the variable \(n\), we have \(n^1\) in the first term and \(n^2\) in the second term. The lowest power of \(n\) present is \(n^1\), or simply \(n\).
Finally, we combine the GCF of the coefficients and the variables. Multiplying the GCF of the coefficients (3) by the GCF of the variables (\(m\) and \(n\)), we get:
\[3 \times m \times n = 3mn.\]
Comparing this result to the given options:
We can see that the correct option is (a).
In manufacturing, finding the GCF is like determining the largest common batch size to produce different types of goods with shared resources. For example, if a factory makes two types of products requiring \(3m^2n\) and \(12mn^2\) units of a specific raw material respectively, the GCF \(3mn\) represents the largest common quantity of base components that can be extracted or optimized from both production lines. Simplifying such expressions helps engineers and managers identify efficiencies and streamline their resource management.