To determine the degree of the polynomial [tex]\( 3x^4 y - 6xy^7 + 2x^4 y^8 \)[/tex], we need to follow these steps:
### Step 1: Identify the degree of each term
1. First term: [tex]\( 3x^4 y \)[/tex]
- The term [tex]\( 3x^4 y \)[/tex] consists of [tex]\( x^4 \)[/tex] and [tex]\( y \)[/tex].
- The degrees of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this term are 4 and 1, respectively.
- The total degree of this term is [tex]\( 4 + 1 = 5 \)[/tex].
2. Second term: [tex]\( -6xy^7 \)[/tex]
- The term [tex]\( -6xy^7 \)[/tex] consists of [tex]\( x \)[/tex] and [tex]\( y^7 \)[/tex].
- The degrees of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this term are 1 and 7, respectively.
- The total degree of this term is [tex]\( 1 + 7 = 8 \)[/tex].
3. Third term: [tex]\( 2x^4 y^8 \)[/tex]
- The term [tex]\( 2x^4 y^8 \)[/tex] consists of [tex]\( x^4 \)[/tex] and [tex]\( y^8 \)[/tex].
- The degrees of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this term are 4 and 8, respectively.
- The total degree of this term is [tex]\( 4 + 8 = 12 \)[/tex].
### Step 2: Determine the degree of the polynomial
The degree of a polynomial is the highest degree of its terms.
- The degrees of the individual terms are 5, 8, and 12.
- The highest degree among these is 12.
Therefore, the degree of the polynomial [tex]\( 3x^4 y - 6xy^7 + 2x^4 y^8 \)[/tex] is [tex]\( \boxed{12} \)[/tex].
