To determine the degree of the polynomial [tex]\( -u y^4 - 6 y^6 w^2 u^2 + 2 w^{11} + 5 \)[/tex], we need to examine each term’s degree and then find the highest degree among them.
1. First term: [tex]\(-u y^4\)[/tex]:
- The degree of [tex]\(y\)[/tex] is 4.
- The degree of [tex]\(u\)[/tex] is 1.
- Since there is no [tex]\(w\)[/tex] in this term, its degree is 0.
- Thus, the total degree of this term is [tex]\(4\)[/tex].
2. Second term: [tex]\(-6 y^6 w^2 u^2\)[/tex]:
- The degree of [tex]\(y\)[/tex] is 6.
- The degree of [tex]\(w\)[/tex] is 2.
- The degree of [tex]\(u\)[/tex] is 2.
- The total degree of this term is [tex]\(6 + 2 + 2 = 10\)[/tex].
3. Third term: [tex]\(2 w^{11}\)[/tex]:
- The degree of [tex]\(w\)[/tex] is 11.
- Since there is no [tex]\(y\)[/tex] or [tex]\(u\)[/tex] in this term, their degrees are 0.
- Thus, the total degree of this term is 11.
4. Fourth term: [tex]\(5\)[/tex]:
- This is a constant term.
- A constant term has a degree of 0.
Now that we have analyzed each term’s degree:
- The degrees of the terms are: 4, 10, 11, and 0.
The degree of the polynomial is the highest degree among its terms, which in this case is [tex]\(11\)[/tex].
Therefore, the degree of the polynomial [tex]\( -u y^4 - 6 y^6 w^2 u^2 + 2 w^{11} + 5 \)[/tex] is [tex]\(11\)[/tex].
