Question: Factor the polynomial. \[ 4m^5 n^4 - 36m^4 n^3 \] Select the correct choice below and, if…

Factor the polynomial.

\[ 4m^5 n^4 - 36m^4 n^3 \]

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

A. \[ 4m^5 n^4 - 36m^4 n^3 = \boxed{\phantom{0}} \] (Factor completely. Simplify your answer.)

B. The polynomial is prime.

Solution

To factor the polynomial \(4m^5n^4 - 36m^4n^3\), follow these steps: Identify the greatest common factor (GCF) of the terms: The coefficients 4 and 36 have a GCF of 4. For the variables \(m\) and \(n\), we look for the lowest powers in each term: - For \(m\): \(m^4\) - For \(n\): \(n^3\) The GCF is \(4m^4n^3\). Factor out the GCF from the polynomial: \[ 4m^5n^4 - 36m^4n^3 = 4m^4n^3(mn - 9) \] Check if the polynomial inside the parentheses can be factored further: \(mn - 9\) cannot be factored further with integer coefficients, so the factored polynomial is: \[ 4m^4n^3(mn - 9) \] The polynomial is not prime, so option A is correct.