Let f(x) polynomial of degree 5 with leading coefficient unity such that f(1)=5, f(2)=4,f(3)=3,f(4)=2,f(5)=1, then f(6) is equal to (a).0 (b). 24 (c). 120 (d). 720

Options

A. 0
B. 24
C. 120
D. 720

Answer

To solve the problem, we need to find the value of \( f(6) \) for the polynomial \( f(x) \) of degree 5 with leading coefficient 1, given the values of \( f(1) = 5 \), \( f(2) = 4 \), \( f(3) = 3 \), \( f(4) = 2 \), and \( f(5) = 1 \).

### Step-by-Step Solution:

1. **Understanding the Polynomial**:
Since \( f(x) \) is a polynomial of degree 5 with leading coefficient 1, we can express it in the form:
\[ f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \]
However, we will use a different approach to find \( f(6) \) based on the given values.

2. **Constructing a New Function**:
Let's define a new function \( g(x) \) as follows:
\[ g(x) = f(x) - (6 - x) \]
This means that \( g(x) \) will also be a polynomial of degree 5, and we can analyze its roots based on the given values of \( f(x) \).

3. **Finding Values of \( g(x) \)**:
Now, we can evaluate \( g(x) \) at the given points:
- \( g(1) = f(1) - (6 - 1) = 5 - 5 = 0 \)
- \( g(2) = f(2) - (6 - 2) = 4 - 4 = 0 \)
- \( g(3) = f(3) - (6 - 3) = 3 - 3 = 0 \)
- \( g(4) = f(4) - (6 - 4) = 2 - 2 = 0 \)
- \( g(5) = f(5) - (6 - 5) = 1 - 1 = 0 \)

Thus, \( g(x) \) has roots at \( x = 1, 2, 3, 4, 5 \).

4. **Expressing \( g(x) \)**:
Since \( g(x) \) is a degree 5 polynomial with roots at 1, 2, 3, 4, and 5, we can write:
\[ g(x) = k(x - 1)(x - 2)(x - 3)(x - 4)(x - 5) \]
where \( k \) is a constant. Since \( f(x) \) has a leading coefficient of 1, we know that \( k \) must be 1. Therefore:
\[ g(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5) \]

5. **Finding \( f(6) \)**:
Now we can find \( f(6) \):
\[ f(6) = g(6) + (6 - 6) = g(6) \]
We calculate \( g(6) \):
\[ g(6) = (6 - 1)(6 - 2)(6 - 3)(6 - 4)(6 - 5) = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

6. **Conclusion**:
Therefore, the value of \( f(6) \) is:
\[ f(6) = 120 \]

### Final Answer:
The value of \( f(6) \) is \( \boxed{120} \).