Questions: Factor completely. 6 m^6 n + 7 m^5 n^2 + 2 m^4 n^3 Select the correct choice below and fill in any answer boxes within your choice. A. 6 m^6 n + 7 m^5 n^2 + 2 m^4 n^3 = B. The trinomial is not factorable.

Factor completely.
6 m^6 n + 7 m^5 n^2 + 2 m^4 n^3

Select the correct choice below and fill in any answer boxes within your choice.
A. 6 m^6 n + 7 m^5 n^2 + 2 m^4 n^3 =
B. The trinomial is not factorable.

Transcript text: Factor completely. \[ 6 m^{6} n+7 m^{5} n^{2}+2 m^{4} n^{3} \] Select the correct choice below and fill in any answer boxes within your choice. A. $6 m^{6} n+7 m^{5} n^{2}+2 m^{4} n^{3}=$ $\square$ B. The trinomial is not factorable.

Solution

Solution Steps

To factor the given expression completely, we first look for the greatest common factor (GCF) of all the terms. Each term contains powers of $m$ and $n$, so we find the lowest power of each variable present in all terms. After factoring out the GCF, we check if the remaining polynomial can be factored further.

Step 1: Identify the Expression

We start with the expression: \[ 6 m^{6} n + 7 m^{5} n^{2} + 2 m^{4} n^{3} \]

Step 2: Factor Out the Greatest Common Factor

The greatest common factor (GCF) of the terms is $m^{4} n$. We factor this out: \[ m^{4} n (6 m^{2} + 7 m n + 2 n^{2}) \]

Step 3: Factor the Remaining Polynomial

Next, we need to factor the quadratic polynomial $6 m^{2} + 7 m n + 2 n^{2}$. This factors into: \[ (2 m + n)(3 m + 2 n) \]

Step 4: Combine the Factors

Combining the factors, we have: \[ 6 m^{6} n + 7 m^{5} n^{2} + 2 m^{4} n^{3} = m^{4} n (2 m + n)(3 m + 2 n) \]