`p(t)=t^(5)-3t^(4)-kt+7t^(2)`
In the polynomial function above, k is a nonzero constant. If p(t) is divisible by t-3, what is the value of k?

Answer

To solve the problem, we need to determine the value of \( k \) such that the polynomial \( p(t) = t^5 - 3t^4 - kt + 7t^2 \) is divisible by \( t - 3 \). According to the Factor Theorem, if \( p(t) \) is divisible by \( t - 3 \), then \( p(3) = 0 \).

### Step-by-Step Solution:

1. **Substitute \( t = 3 \) into the polynomial \( p(t) \)**:
\[ p(3) = 3^5 - 3 \cdot 3^4 - k \cdot 3 + 7 \cdot 3^2 \]

2. **Calculate \( 3^5 \), \( 3^4 \), and \( 3^2 \)**:
- \( 3^5 = 243 \)
- \( 3^4 = 81 \)
- \( 3^2 = 9 \)

3. **Substitute these values into the polynomial**:
\[ p(3) = 243 - 3 \cdot 81 - 3k + 7 \cdot 9 \]

4. **Calculate \( 3 \cdot 81 \) and \( 7 \cdot 9 \)**:
- \( 3 \cdot 81 = 243 \)
- \( 7 \cdot 9 = 63 \)

5. **Now substitute these results into the equation**:
\[ p(3) = 243 - 243 - 3k + 63 \]

6. **Simplify the equation**:
\[ p(3) = 0 - 3k + 63 = 63 - 3k \]

7. **Set \( p(3) \) equal to 0** (since \( p(t) \) is divisible by \( t - 3 \)):
\[ 63 - 3k = 0 \]

8. **Solve for \( k \)**:
\[ 3k = 63 \\ k = \frac{63}{3} = 21 \]

Thus, the value of \( k \) is \( \boxed{21} \).