Certainly! Let's go through the solution step-by-step:
We start with the polynomial:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]
Step 1: Simplify the Polynomial by Combining Like Terms
First, identify the like terms in the polynomial. We see that [tex]\(-2 m^2 n^3\)[/tex] and [tex]\(2 m^2 n^3\)[/tex] are like terms because they have the same powers of [tex]\(m\)[/tex] and [tex]\(n\)[/tex].
Combining these terms, we get:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 = 0 \][/tex]
So, these two terms cancel each other out.
Step 2: Identify the Remaining Terms
After canceling the like terms, we are left with:
[tex]\[ 7 n^2 - 6 m^4 \][/tex]
Step 3: Determine the Degree of the Polynomial
We need to check the degree of each term:
- The term [tex]\(7 n^2\)[/tex] has a degree of 2 because the exponent of [tex]\(n\)[/tex] is 2.
- The term [tex]\(-6 m^4\)[/tex] has a degree of 4 because the exponent of [tex]\(m\)[/tex] is 4.
Since [tex]\(-6 m^4\)[/tex] is of degree 4, and [tex]\(7 n^2\)[/tex] is of lower degree, the overall degree of the polynomial, defined by the highest power, is 4.
Step 4: Confirm the Polynomial is a Binomial of Degree 4
A binomial consists of exactly two distinct terms. The simplified polynomial, [tex]\(7 n^2 - 6 m^4\)[/tex], has two distinct terms:
[tex]\[ 7 n^2 \quad \text{and} \quad -6 m^4 \][/tex]
Thus, it is a binomial.
For the polynomial to be a binomial of degree 4, we need to check the missing exponent on the [tex]\(m\)[/tex] in the initial term [tex]\(2 m^x n^3\)[/tex] to ensure that it forms a matching term with degree observations and combines correctly. The exponent [tex]\(x\)[/tex] on [tex]\(m\)[/tex] in [tex]\(m^x n^3\)[/tex] should make it equal degree term contributing correctly for the simplifying steps:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 \][/tex]
Thus, x must have been 2 (since simplifying these terms correctly, cancel each other out if combining correctly into 0).
Thus, the answer is:
[tex]\[ \boxed{2} \][/tex]
