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 \( b \) in the function \( f(x) = ax^3 + x^2 + bx + c \) such that the graph of the function is symmetric about the line \( x = 2 \).

### Step 1: Understanding Symmetry
A function is symmetric about a vertical line \( x = k \) if \( f(k - x) = f(k + x) \). In this case, \( k = 2 \). Therefore, we need to check if \( f(2 - x) = f(2 + x) \).

### Step 2: Calculate \( f(2 - x) \)
Substituting \( 2 - x \) into the function:
\[ f(2 - x) = a(2 - x)^3 + (2 - x)^2 + b(2 - x) + c \]
Expanding this:
\[ = a(8 - 12x + 6x^2 - x^3) + (4 - 4x + x^2) + b(2 - x) + c \]
\[ = 8a - 12ax + 6ax^2 - ax^3 + 4 - 4x + x^2 + 2b - bx + c \]
Combining like terms:
\[ = (-a)x^3 + (6a + 1)x^2 + (-12a - 4 - b)x + (8a + 4 + 2b + c) \]

### Step 3: Calculate \( f(2 + x) \)
Now substituting \( 2 + x \) into the function:
\[ f(2 + x) = a(2 + x)^3 + (2 + x)^2 + b(2 + x) + c \]
Expanding this:
\[ = a(8 + 12x + 6x^2 + x^3) + (4 + 4x + x^2) + b(2 + x) + c \]
\[ = 8a + 12ax + 6ax^2 + ax^3 + 4 + 4x + x^2 + 2b + bx + c \]
Combining like terms:
\[ = ax^3 + (6a + 1)x^2 + (12a + 4 + b)x + (8a + 4 + 2b + c) \]

### Step 4: Set the Two Expressions Equal
Since \( f(2 - x) = f(2 + x) \), we equate the coefficients from both expanded forms:
1. Coefficient of \( x^3 \):
\[ -a = a \implies 2a = 0 \implies a = 0 \]

2. Coefficient of \( x^2 \):
\[ 6a + 1 = 6(0) + 1 = 1 \quad \text{(this is always true)} \]

3. Coefficient of \( x \):
\[ -12a - 4 - b = 12a + 4 + b \]
Substituting \( a = 0 \):
\[ -4 - b = 4 + b \implies -4 - 4 = 2b \implies -8 = 2b \implies b = -4 \]

### Step 5: Calculate \( a + b \)
Now we have:
\[ a = 0, \quad b = -4 \]
Thus,
\[ a + b = 0 - 4 = -4 \]

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