If the garph of the function `f(x)=ax^(3)+x^(2)+bx+c` is symmetric about the line x = 2, then the value of `a+b` is equal to

Options

A. 10
B. `-4`
C. 16
D. `-10`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the function \( f(x) = ax^3 + x^2 + bx + c \) is symmetric about the line \( x = 2 \).

### Step 1: Understand the symmetry condition
For a function to be symmetric about the line \( x = 2 \), it must satisfy the condition:
\[ f(2 + x) = f(2 - x) \]
for all \( x \).

### Step 2: Calculate \( f(2 + x) \) and \( f(2 - x) \)
First, we compute \( f(2 + x) \):
\[ f(2 + x) = a(2 + x)^3 + (2 + x)^2 + b(2 + x) + c \]
Expanding this:
\[ = a(8 + 12x + 6x^2 + x^3) + (4 + 4x + x^2) + b(2 + x) + c \]
\[ = 8a + 4 + 2b + c + (12a + 4 + b)x + (6a + 1)x^2 + ax^3 \]

Now, we compute \( f(2 - x) \):
\[ f(2 - x) = a(2 - x)^3 + (2 - x)^2 + b(2 - x) + c \]
Expanding this:
\[ = a(8 - 12x + 6x^2 - x^3) + (4 - 4x + x^2) + b(2 - x) + c \]
\[ = 8a + 4 + 2b + c + (-12a - 4 - b)x + (6a + 1)x^2 - ax^3 \]

### Step 3: Set the two expressions equal
Setting \( f(2 + x) = f(2 - x) \):
\[ 8a + 4 + 2b + c + (12a + 4 + b)x + (6a + 1)x^2 + ax^3 = 8a + 4 + 2b + c + (-12a - 4 - b)x + (6a + 1)x^2 - ax^3 \]

### Step 4: Equate coefficients
From the equality, we can equate coefficients of like powers of \( x \):
1. Coefficient of \( x^3 \): \( a = -a \) implies \( 2a = 0 \) so \( a = 0 \).
2. Coefficient of \( x^2 \): \( 6a + 1 = 6a + 1 \) (this is always true).
3. Coefficient of \( x \): \( 12a + 4 + b = -12a - 4 - b \).
Substituting \( a = 0 \):
\[ 4 + b = -4 - b \implies 2b = -8 \implies b = -4. \]

### Step 5: Find \( a + b \)
Now we have \( a = 0 \) and \( b = -4 \). Thus:
\[ a + b = 0 + (-4) = -4. \]

### Final Answer
The value of \( a + b \) is \( \boxed{-4} \).