Certainly! To determine the degree of a polynomial, we look at each term separately and find the sum of the exponents for each variable within a term. The polynomial given is:
[tex]\[ 13x^7yz - 7x^5y^2 + x^4yz^3 \][/tex]
Let's break this down term by term:
1. First term: [tex]\( 13x^7yz \)[/tex]
- The degree of [tex]\( x \)[/tex] is 7.
- The degree of [tex]\( y \)[/tex] is 1.
- The degree of [tex]\( z \)[/tex] is 1.
- The sum of these degrees is [tex]\( 7 + 1 + 1 = 9 \)[/tex].
2. Second term: [tex]\( -7x^5y^2 \)[/tex]
- The degree of [tex]\( x \)[/tex] is 5.
- The degree of [tex]\( y \)[/tex] is 2.
- The degree of [tex]\( z \)[/tex] is 0 (since [tex]\( z \)[/tex] does not appear in this term).
- The sum of these degrees is [tex]\( 5 + 2 = 7 \)[/tex].
3. Third term: [tex]\( x^4yz^3 \)[/tex]
- The degree of [tex]\( x \)[/tex] is 4.
- The degree of [tex]\( y \)[/tex] is 1.
- The degree of [tex]\( z \)[/tex] is 3.
- The sum of these degrees is [tex]\( 4 + 1 + 3 = 8 \)[/tex].
Now, we need to find the degree of the entire polynomial, which is the highest degree among all its terms. Thus, we compare the degrees calculated:
- Degree of [tex]\( 13x^7yz \)[/tex] is 9.
- Degree of [tex]\( -7x^5y^2 \)[/tex] is 7.
- Degree of [tex]\( x^4yz^3 \)[/tex] is 8.
The highest degree is 9.
Therefore, the degree of the polynomial [tex]\( 13x^7yz - 7x^5y^2 + x^4yz^3 \)[/tex] is [tex]\( 9 \)[/tex].
