Question: Factor the polynomial. \[ 4m^4 n^3 - 32m^3 n^2 \] Select the correct choice below and, if…

Factor the polynomial.

\[ 4m^4 n^3 - 32m^3 n^2 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \[ 4m^4 n^3 - 32m^3 n^2 = \boxed{\phantom{0}} \] (Factor completely. Simplify your answer.)

B. The polynomial is prime.

Solution

The problem is to factor the polynomial \(4m^4n - 32m^3n^2\). First, identify the greatest common factor (GCF) of the terms. The GCF of \(4m^4n\) and \(32m^3n^2\) is \(4m^3n\). Factor out the GCF from the polynomial: \[ 4m^4n - 32m^3n^2 = 4m^3n(m - 8n) \] So the completely factored form of the polynomial is: \[ 4m^3n(m - 8n) \]