To determine the correct statement about the given polynomial [tex]\(-3 x^4 y^3 + 8 x y^5 - 3 + 18 x^3 y^4 - 3 x y^5\)[/tex], we need to simplify it and analyze its terms and degree.
### Step 1: Combine Like Terms
First, combine any like terms in the polynomial. Like terms are those that have the same variables raised to the same powers.
The given polynomial is:
[tex]\[
-3 x^4 y^3 + 8 x y^5 - 3 + 18 x^3 y^4 - 3 x y^5
\][/tex]
Combine the terms [tex]\(8 x y^5\)[/tex] and [tex]\(-3 x y^5\)[/tex]:
[tex]\[
8 x y^5 - 3 x y^5 = 5 x y^5
\][/tex]
Now, rewrite the polynomial with combined like terms:
[tex]\[
-3 x^4 y^3 + 5 x y^5 - 3 + 18 x^3 y^4
\][/tex]
### Step 2: Identify the Number of Unique Terms
After combining like terms, we have:
[tex]\[
-3 x^4 y^3, \quad 5 x y^5, \quad -3, \quad \text{and}, \quad 18 x^3 y^4
\][/tex]
We see that there are 4 unique terms here.
### Step 3: Determine the Degree of the Polynomial
The degree of a polynomial is the highest degree of any term within the polynomial. The degree of a term is the sum of the exponents of the variables in that term.
- For [tex]\(-3 x^4 y^3\)[/tex]:
[tex]\[ \text{Degree} = 4 + 3 = 7 \][/tex]
- For [tex]\(5 x y^5\)[/tex]:
[tex]\[ \text{Degree} = 1 + 5 = 6 \][/tex]
- For the constant term [tex]\(-3\)[/tex]:
[tex]\[ \text{Degree} = 0 \][/tex]
- For [tex]\(18 x^3 y^4\)[/tex]:
[tex]\[ \text{Degree} = 3 + 4 = 7 \][/tex]
The highest degree among these is [tex]\(7\)[/tex], hence the degree of the polynomial is [tex]\(7\)[/tex].
### Conclusion
The polynomial after simplification has 4 terms and a degree of 7. Therefore, the correct statement is:
"It has 4 terms and a degree of 7".
