To find the degree and the leading coefficient of the polynomial, let's analyze the given polynomial:
[tex]\[ -15 v^3 - 23 - 3 v + v^7 \][/tex]
### Step 1: Identify the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable [tex]\( v \)[/tex] present in the polynomial. In the given polynomial, the terms are:
- [tex]\( -15 v^3 \)[/tex] which has a degree of 3
- [tex]\( -23 \)[/tex] which can be considered as [tex]\( -23 v^0 \)[/tex] and has a degree of 0
- [tex]\( -3 v \)[/tex] which has a degree of 1
- [tex]\( v^7 \)[/tex] which has a degree of 7
The highest power among these is [tex]\( v^7 \)[/tex]. Therefore, the degree of the polynomial is [tex]\( 7 \)[/tex].
### Step 2: Identify the Leading Coefficient
The leading coefficient is the coefficient of the term with the highest degree. From the previous step, we identified that the term with the highest degree is [tex]\( v^7 \)[/tex]. The coefficient of [tex]\( v^7 \)[/tex] is [tex]\( 1 \)[/tex].
### Conclusion
- The degree of the polynomial is [tex]\( 7 \)[/tex].
- The leading coefficient is [tex]\( 1 \)[/tex].
Hence, the degree and leading coefficient of the polynomial [tex]\( -15 v^3 - 23 - 3 v + v^7 \)[/tex] are:
[tex]\[ \boxed{(7, 1)} \][/tex]
