To determine the missing exponent [tex]\( x \)[/tex] on the [tex]\( m \)[/tex] in the second term of the polynomial [tex]\( -2m^2 n^3 + 2m^x n^3 + 7n^2 - 6m^4 \)[/tex] so that the polynomial simplifies to a binomial of degree 4, follow these steps:
1. Identify Like Terms:
The terms [tex]\( -2m^2 n^3 \)[/tex] and [tex]\( 2m^x n^3 \)[/tex] are like terms if their exponents on [tex]\( m \)[/tex] are the same.
2. Combine Like Terms:
For the polynomial to simplify correctly, we need the terms [tex]\( -2m^2 n^3 \)[/tex] and [tex]\( 2m^x n^3 \)[/tex] to either cancel each other out or to leave us with a simplified form.
3. Set Exponents Equal:
To simplify, the exponents on [tex]\( m \)[/tex] in both terms must be the same. So we set the exponents equal:
[tex]\[
2 = x
\][/tex]
This implies that [tex]\( x \)[/tex] must be 2.
4. Simplifying the Polynomial:
When [tex]\( x = 2 \)[/tex]:
[tex]\[
-2m^2 n^3 + 2m^2 n^3 + 7n^2 - 6m^4
\][/tex]
The terms [tex]\( -2m^2 n^3 \)[/tex] and [tex]\( 2m^2 n^3 \)[/tex] cancel each other out:
[tex]\[
-2m^2 n^3 + 2m^2 n^3 = 0
\][/tex]
5. Simplified Polynomial:
So the polynomial simplifies to:
[tex]\[
0 + 7n^2 - 6m^4 = 7n^2 - 6m^4
\][/tex]
The remaining terms are [tex]\( 7n^2 \)[/tex] and [tex]\( -6m^4 \)[/tex].
6. Determine the Degree of the Polynomial:
A binomial is an algebraic expression consisting of exactly two terms:
[tex]\[
7n^2 - 6m^4
\][/tex]
The degree of the polynomial is given by the highest degree of its terms. Here, the term [tex]\( -6m^4 \)[/tex] determines the degree, which is 4.
Therefore, the missing exponent [tex]\( x \)[/tex] on the [tex]\( m \)[/tex] in the second term must be:
[tex]\[
\boxed{2}
\][/tex]
