If the degree of the polynomial `(p^(6)+(3)/(7))(p^(n)+3p)` is 9, then the value of n is

Options

A. 1
B. 3
C. 6
D. 18

Answer

To solve the problem, we need to find the value of \( n \) in the polynomial \( (p^6 + \frac{3}{7})(p^n + 3p) \) given that the degree of the polynomial is 9.

### Step-by-step Solution:

1. **Identify the degrees of the individual terms**:
- The first part of the polynomial is \( p^6 \), which has a degree of 6.
- The second part of the polynomial is \( p^n + 3p \). The degree of \( p^n \) is \( n \) and the degree of \( 3p \) is 1. Therefore, the degree of \( p^n + 3p \) is \( \max(n, 1) \).

2. **Determine the overall degree of the polynomial**:
- The degree of the product of two polynomials is the sum of their degrees. Thus, the degree of the polynomial \( (p^6 + \frac{3}{7})(p^n + 3p) \) is:
\[ \text{Degree} = 6 + \max(n, 1) \]

3. **Set the degree equal to 9**:
- According to the problem, the degree of the polynomial is given as 9. Therefore, we can set up the equation:
\[ 6 + \max(n, 1) = 9 \]

4. **Solve for \( n \)**:
- Rearranging the equation gives:
\[ \max(n, 1) = 9 - 6 \]
\[ \max(n, 1) = 3 \]
- This means either \( n = 3 \) or \( n < 1 \) (which is not possible since \( n \) must be a non-negative integer). Therefore, we conclude:
\[ n = 3 \]

### Final Answer:
The value of \( n \) is \( 3 \).