To solve for the degree of the polynomial [tex]\( x^2 y + 3 x^3 y^2 + y^4 \)[/tex], we need to determine the degree of each monomial term and find the highest degree among them.
1. First term: [tex]\( x^2 y \)[/tex]
- The powers of the variables are: [tex]\( x^2 \)[/tex] and [tex]\( y \)[/tex].
- Degree of this term is [tex]\( 2 \)[/tex] (from [tex]\( x^2 \)[/tex]) + [tex]\( 1 \)[/tex] (from [tex]\( y \)[/tex]) = [tex]\( 3 \)[/tex].
2. Second term: [tex]\( 3 x^3 y^2 \)[/tex]
- The powers of the variables are: [tex]\( x^3 \)[/tex] and [tex]\( y^2 \)[/tex].
- Degree of this term is [tex]\( 3 \)[/tex] (from [tex]\( x^3 \)[/tex]) + [tex]\( 2 \)[/tex] (from [tex]\( y^2 \)[/tex]) = [tex]\( 5 \)[/tex].
3. Third term: [tex]\( y^4 \)[/tex]
- The power of the variable [tex]\( y \)[/tex] is: [tex]\( 4 \)[/tex].
- Degree of this term is [tex]\( 0 \)[/tex] (as [tex]\( x \)[/tex] is absent) + [tex]\( 4 \)[/tex] (from [tex]\( y^4 \)[/tex]) = [tex]\( 4 \)[/tex].
The degree of the polynomial is the highest degree among these monomial terms. Therefore, comparing the degrees:
- [tex]\( x^2 y \)[/tex] has a degree of [tex]\( 3 \)[/tex].
- [tex]\( 3 x^3 y^2 \)[/tex] has a degree of [tex]\( 5 \)[/tex].
- [tex]\( y^4 \)[/tex] has a degree of [tex]\( 4 \)[/tex].
The highest degree is [tex]\( 5 \)[/tex].
Thus, the degree of the polynomial [tex]\( x^2 y + 3 x^3 y^2 + y^4 \)[/tex] is [tex]\( 5 \)[/tex].
The correct answer is:
```
5
```
