To solve the problem, let's go through the polynomial step-by-step and simplify it.
Given polynomial:
[tex]\[ -10 m^4 n^3 + 8 m^2 n^6 + 3 m^4 n^3 - 2 m^2 n^6 - 6 m^2 n^6 \][/tex]
First, we'll combine like terms. We have terms with [tex]\(m^4 n^3\)[/tex] and terms with [tex]\(m^2 n^6\)[/tex].
1. Combine the [tex]\(m^4 n^3\)[/tex] terms:
[tex]\[ -10 m^4 n^3 + 3 m^4 n^3 = (-10 + 3) m^4 n^3 = -7 m^4 n^3 \][/tex]
2. Combine the [tex]\(m^2 n^6\)[/tex] terms:
[tex]\[ 8 m^2 n^6 - 2 m^2 n^6 - 6 m^2 n^6 = (8 - 2 - 6) m^2 n^6 = 0 m^2 n^6 \][/tex]
Since the result is 0, this entire term vanishes.
So, the simplified polynomial is:
[tex]\[ -7 m^4 n^3 \][/tex]
Now, let's analyze the characteristics of the polynomial:
- It only consists of one term, which makes it a monomial.
- The degree of the polynomial is the sum of the exponents in the monomial:
[tex]\[ \text{Degree} = 4 (exponent\ of\ m) + 3 (exponent\ of\ n) = 4 + 3 = 7 \][/tex]
Thus, the correct statement about the polynomial after it has been fully simplified is:
It is a monomial with a degree of 7.
