To tackle the problem, let's analyze and simplify the given polynomial step-by-step:
The polynomial given is [tex]\(-3 x^4 y^3 + 8 x y^5 - 3 + 18 x^3 y^4 - 3 x y^5\)[/tex].
First, let's combine like terms:
1. Identify the terms:
- [tex]\(-3 x^4 y^3\)[/tex]
- [tex]\(8 x y^5\)[/tex]
- [tex]\(-3\)[/tex]
- [tex]\(18 x^3 y^4\)[/tex]
- [tex]\(-3 x y^5\)[/tex]
2. Combine the like 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]
So, the polynomial is now:
[tex]\[-3 x^4 y^3 + 5 x y^5 - 3 + 18 x^3 y^4\][/tex]
Next, count the number of terms and find the degree of the polynomial:
1. Count the number of terms:
- [tex]\(-3 x^4 y^3\)[/tex]
- [tex]\(5 x y^5\)[/tex]
- [tex]\(-3\)[/tex]
- [tex]\(18 x^3 y^4\)[/tex]
There are 4 distinct terms.
2. Find the degree of the polynomial:
- The degree of [tex]\(-3 x^4 y^3\)[/tex] is [tex]\(4+3 = 7\)[/tex].
- The degree of [tex]\(5 x y^5\)[/tex] is [tex]\(1+5 = 6\)[/tex].
- The degree of [tex]\(-3\)[/tex] is [tex]\(0\)[/tex].
- The degree of [tex]\(18 x^3 y^4\)[/tex] is [tex]\(3+4 = 7\)[/tex].
The highest degree among the terms is 7.
Therefore, the polynomial simplifies to four distinct terms with the highest degree being 7. Thus, the correct statement is:
It has 4 terms and a degree of 7.
