If f(x) is a polynominal of degree 4 with leading coefficient '1' satisfying f(1)=10,f(2)=20 and f(3)=30, then `((f(12)+f(-8))/(19840))` is …………. .

Answer

To solve the problem, we need to find the polynomial \( f(x) \) of degree 4 with leading coefficient 1 that satisfies the given conditions. Let's go through the solution step by step.

### Step 1: Define the Polynomial
Since \( f(x) \) is a polynomial of degree 4 with leading coefficient 1, we can express it as:
\[ f(x) = x^4 + ax^3 + bx^2 + cx + d \]
where \( a, b, c, d \) are constants we need to determine.

### Step 2: Use Given Conditions
We know:
1. \( f(1) = 10 \)
2. \( f(2) = 20 \)
3. \( f(3) = 30 \)

We can set up equations based on these conditions.

### Step 3: Set Up the Equations
1. For \( f(1) = 10 \):
\[ 1^4 + a(1^3) + b(1^2) + c(1) + d = 10 \implies 1 + a + b + c + d = 10 \implies a + b + c + d = 9 \quad \text{(Equation 1)} \]

2. For \( f(2) = 20 \):
\[ 2^4 + a(2^3) + b(2^2) + c(2) + d = 20 \implies 16 + 8a + 4b + 2c + d = 20 \implies 8a + 4b + 2c + d = 4 \quad \text{(Equation 2)} \]

3. For \( f(3) = 30 \):
\[ 3^4 + a(3^3) + b(3^2) + c(3) + d = 30 \implies 81 + 27a + 9b + 3c + d = 30 \implies 27a + 9b + 3c + d = -51 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have a system of three equations:
1. \( a + b + c + d = 9 \)
2. \( 8a + 4b + 2c + d = 4 \)
3. \( 27a + 9b + 3c + d = -51 \)

We can eliminate \( d \) by subtracting Equation 1 from the other two equations.

#### Subtract Equation 1 from Equation 2:
\[ (8a + 4b + 2c + d) - (a + b + c + d) = 4 - 9 \]
This simplifies to:
\[ 7a + 3b + c = -5 \quad \text{(Equation 4)} \]

#### Subtract Equation 1 from Equation 3:
\[ (27a + 9b + 3c + d) - (a + b + c + d) = -51 - 9 \]
This simplifies to:
\[ 26a + 8b + 2c = -60 \quad \text{(Equation 5)} \]

### Step 5: Solve Equations 4 and 5
Now we have:
1. \( 7a + 3b + c = -5 \) (Equation 4)
2. \( 26a + 8b + 2c = -60 \) (Equation 5)

From Equation 4, we can express \( c \):
\[ c = -5 - 7a - 3b \]
Substituting \( c \) into Equation 5:
\[ 26a + 8b + 2(-5 - 7a - 3b) = -60 \]
This simplifies to:
\[ 26a + 8b - 10 - 14a - 6b = -60 \]
\[ 12a + 2b - 10 = -60 \implies 12a + 2b = -50 \implies 6a + b = -25 \quad \text{(Equation 6)} \]

### Step 6: Substitute Back to Find \( a, b, c, d \)
Now we can express \( b \) in terms of \( a \):
\[ b = -25 - 6a \]
Substituting \( b \) into Equation 4:
\[ 7a + 3(-25 - 6a) + c = -5 \]
This gives:
\[ 7a - 75 - 18a + c = -5 \implies -11a + c = 70 \implies c = 70 + 11a \]
Now substituting \( b \) and \( c \) back into Equation 1 to find \( d \):
\[ a + (-25 - 6a) + (70 + 11a) + d = 9 \]
This simplifies to:
\[ a - 6a + 11a + 45 + d = 9 \implies 6a + d = -36 \implies d = -36 - 6a \]

### Step 7: Choose a Value for \( a \)
To find specific values, we can choose \( a = 0 \) (as a simple case):
- \( a = 0 \)
- \( b = -25 \)
- \( c = 70 \)
- \( d = -36 \)

Thus, we have:
\[ f(x) = x^4 - 25x^2 + 70x - 36 \]

### Step 8: Calculate \( f(12) \) and \( f(-8) \)
Now we need to calculate \( f(12) \) and \( f(-8) \):
1. \( f(12) = 12^4 - 25(12^2) + 70(12) - 36 = 20736 - 3600 + 840 - 36 = 17500 \)
2. \( f(-8) = (-8)^4 - 25(-8)^2 + 70(-8) - 36 = 4096 - 1600 - 560 - 36 = 1950 \)

### Step 9: Find the Final Expression
Now we compute:
\[ \frac{f(12) + f(-8)}{19840} = \frac{17500 + 1950}{19840} = \frac{19450}{19840} \]
Simplifying:
\[ = \frac{1945}{1984} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1945}{1984} \]