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

Options

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

Answer

To solve the problem, we need to find the value of the polynomial \( f(x) \) at \( x = 6 \) given the conditions. Let's break it down step by step.

### Step 1: Understand the given conditions
We know that \( f(x) \) is a polynomial of degree 5 with leading coefficient 1. We are given the following values:
- \( f(1) = 5 \)
- \( f(2) = 4 \)
- \( f(3) = 3 \)
- \( f(4) = 2 \)
- \( f(5) = 1 \)

### Step 2: Identify a new function
To simplify the problem, we can define a new function \( g(x) \) such that:
\[ f(x) = g(x) + (6 - x) \]
This means that \( g(x) \) will capture the deviation of \( f(x) \) from the linear function \( 6 - x \).

### Step 3: Calculate \( g(x) \) at known points
Now, substituting the known values into our new function:
- For \( x = 1 \):
\[ f(1) = g(1) + (6 - 1) = g(1) + 5 \implies g(1) = 0 \]
- For \( x = 2 \):
\[ f(2) = g(2) + (6 - 2) = g(2) + 4 \implies g(2) = 0 \]
- For \( x = 3 \):
\[ f(3) = g(3) + (6 - 3) = g(3) + 3 \implies g(3) = 0 \]
- For \( x = 4 \):
\[ f(4) = g(4) + (6 - 4) = g(4) + 2 \implies g(4) = 0 \]
- For \( x = 5 \):
\[ f(5) = g(5) + (6 - 5) = g(5) + 1 \implies g(5) = 0 \]

### Step 4: Roots of \( g(x) \)
From the calculations above, we see that \( g(x) \) has roots at \( x = 1, 2, 3, 4, 5 \). Therefore, we can express \( g(x) \) as:
\[ g(x) = k(x - 1)(x - 2)(x - 3)(x - 4)(x - 5) \]
where \( k \) is a constant.

### Step 5: Determine the leading coefficient
Since \( f(x) \) is a polynomial of degree 5 with leading coefficient 1, we know that \( g(x) \) must also have a leading coefficient of 1. Thus, \( k = 1 \).

### Step 6: Write the complete function
Now we can write:
\[ f(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5) + (6 - x) \]

### Step 7: Calculate \( f(6) \)
Now we substitute \( x = 6 \) into \( f(x) \):
\[ f(6) = (6 - 1)(6 - 2)(6 - 3)(6 - 4)(6 - 5) + (6 - 6) \]
Calculating each term:
\[ f(6) = (5)(4)(3)(2)(1) + 0 = 120 + 0 = 120 \]

### Final Answer
Thus, \( f(6) = 120 \).

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