To factor the polynomial [tex]\( 2mn + 3m^6n^2 \)[/tex], we need to find the common factors of both terms and factor them out. Let's break this down step by step:
1. Identify the terms of the polynomial:
The given polynomial is [tex]\( 2mn + 3m^6n^2 \)[/tex].
2. Find the greatest common factor (GCF):
- For the coefficients: The GCF of 2 and 3 is 1.
- For the variable [tex]\( m \)[/tex]: The smallest power of [tex]\( m \)[/tex] is [tex]\( m \)[/tex]. Hence, [tex]\( m \)[/tex] is a common factor.
- For the variable [tex]\( n \)[/tex]: The smallest power of [tex]\( n \)[/tex] is [tex]\( n \)[/tex]. Hence, [tex]\( n \)[/tex] is a common factor.
Therefore, the GCF of the polynomial [tex]\( 2mn + 3m^6n^2 \)[/tex] is [tex]\( mn \)[/tex].
3. Factor out the GCF:
We can write each term as the product of [tex]\( mn \)[/tex] and another expression:
- [tex]\( 2mn = mn \cdot 2 \)[/tex]
- [tex]\( 3m^6n^2 = mn \cdot 3m^5n \)[/tex]
4. Rewrite the polynomial:
Factor [tex]\( mn \)[/tex] out from each term:
[tex]\[
2mn + 3m^6n^2 = mn (2) + mn (3m^5n)
\][/tex]
5. Combine the factored form:
Finally, combine the remaining expressions inside the parentheses:
[tex]\[
2mn + 3m^6n^2 = mn (2 + 3m^5n)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\( 2mn + 3m^6n^2 \)[/tex] is:
[tex]\[
\boxed{mn (2 + 3m^5n)}
\][/tex]
