To find the value of [tex]\( k \)[/tex] in the polynomial [tex]\( x + y^2 - 7xy - 6w^kz^2 \)[/tex] given that the degree of the polynomial is 10, we need to determine which term influences the overall degree and work accordingly.
The degree of a polynomial is determined by the highest degree of its individual terms. Let's analyze the degrees of each term in the polynomial [tex]\( x + y^2 - 7xy - 6w^kz^2 \)[/tex]:
1. For the term [tex]\( x \)[/tex]:
- The degree of [tex]\( x \)[/tex] is 1.
2. For the term [tex]\( y^2 \)[/tex]:
- The degree of [tex]\( y^2 \)[/tex] is 2.
3. For the term [tex]\( -7xy \)[/tex]:
- The degree of [tex]\( -7xy \)[/tex] is the sum of the degrees of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which is [tex]\( 1 + 1 = 2 \)[/tex].
4. For the term [tex]\( -6w^kz^2 \)[/tex]:
- The degree of [tex]\( -6w^kz^2 \)[/tex] is the sum of the degrees of [tex]\( w^k \)[/tex] and [tex]\( z^2 \)[/tex], which is [tex]\( k + 2 \)[/tex].
The given information tells us that the polynomial's total degree is 10. To find the value of [tex]\( k \)[/tex], we need to focus on the term that gives us the highest degree. In this context, [tex]\( -6w^kz^2 \)[/tex] appears to be the term that can give us the target degree of 10 through the combination [tex]\( k + 2 \)[/tex].
Set up the equation for the degree based on the term [tex]\( -6w^kz^2 \)[/tex]:
[tex]\[ k + 2 = 10 \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = 10 - 2 \][/tex]
[tex]\[ k = 8 \][/tex]
So, the value of [tex]\( k \)[/tex] that makes the term [tex]\( -6w^kz^2 \)[/tex] have a total degree of 10 is:
[tex]\[ k = 8 \][/tex]
Thus, the correct answer is 8.
