Alright, let's break down the given polynomial step-by-step to determine the missing exponent on the [tex]\( m \)[/tex] in the term [tex]\( 2 m^2 n^3 \)[/tex].
The given polynomial is:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]
First, simplify the polynomial by combining like terms:
1. Combine the terms [tex]\(-2 m^2 n^3\)[/tex] and [tex]\( 2 m^2 n^3\)[/tex]:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 = 0 \][/tex]
2. After combining these terms, we are left with:
[tex]\[ 0 + 7 n^2 - 6 m^4 \][/tex]
[tex]\[ = 7 n^2 - 6 m^4 \][/tex]
Now we have a simplified polynomial:
[tex]\[ 7 n^2 - 6 m^4 \][/tex]
We can see that this simplified polynomial is a binomial (a polynomial with two terms) consisting of [tex]\( 7 n^2 \)[/tex] and [tex]\( -6 m^4 \)[/tex].
Next, let's examine the degrees of each term in the simplified polynomial:
- The degree of [tex]\( 7 n^2 \)[/tex] is 2 (since [tex]\( n^2 \)[/tex] has an exponent of 2).
- The degree of [tex]\( -6 m^4 \)[/tex] is 4 (since [tex]\( m^4 \)[/tex] has an exponent of 4).
To ensure that the entire polynomial has the highest degree of 4 (the degree of [tex]\( -6 m^4 \)[/tex]), we need to determine which exponent on the [tex]\( m \)[/tex] in the term [tex]\( 2 m^2 n^3 \)[/tex] must be adjusted.
By examining the original polynomial before simplification:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]
Given that the simplified polynomial must result in a degree of 4 and identifying the highest degree in the term [tex]\( -6 m^4 \)[/tex], we infer that the missing exponent on the [tex]\( m \)[/tex] term (in [tex]\( 2 m^2 n^3 \)[/tex]) required to create such a default highest degree is:
[tex]\[ \boxed{4} \][/tex]
