If `(x^(4)+24x^(2)+28)/((x^(2)+1)^(3))=A/((x^(2)+1))+B/((x^(2)+1)^(2))+C/((x^(2)+1)^(3))`, then A + C =

Options

A. 6
B. 27
C. 28
D. 29

Answer

To solve the problem step by step, we start with the given equation:

\[ \frac{x^4 + 24x^2 + 28}{(x^2 + 1)^3} = \frac{A}{(x^2 + 1)} + \frac{B}{(x^2 + 1)^2} + \frac{C}{(x^2 + 1)^3} \]

### Step 1: Clear the Denominator
Multiply both sides by \((x^2 + 1)^3\):

\[ x^4 + 24x^2 + 28 = A(x^2 + 1)^2 + B(x^2 + 1) + C \]

### Step 2: Expand the Right Side
Now, we expand the right-hand side:

1. Expand \(A(x^2 + 1)^2\):
\[ A(x^2 + 1)^2 = A(x^4 + 2x^2 + 1) = Ax^4 + 2Ax^2 + A \]

2. Expand \(B(x^2 + 1)\):
\[ B(x^2 + 1) = Bx^2 + B \]

3. Combine these expansions:
\[ Ax^4 + (2A + B)x^2 + (A + B + C) \]

### Step 3: Set Up the Equation
Now we equate the coefficients from both sides:

\[ x^4 + 24x^2 + 28 = Ax^4 + (2A + B)x^2 + (A + B + C) \]

### Step 4: Compare Coefficients
From the equation, we can compare coefficients of \(x^4\), \(x^2\), and the constant term:

1. Coefficient of \(x^4\):
\[ A = 1 \]

2. Coefficient of \(x^2\):
\[ 2A + B = 24 \]

3. Constant term:
\[ A + B + C = 28 \]

### Step 5: Solve for A, B, and C
Now we substitute \(A = 1\) into the equations:

1. From \(A = 1\):
\[ 2(1) + B = 24 \implies 2 + B = 24 \implies B = 22 \]

2. Substitute \(A\) and \(B\) into the constant term equation:
\[ 1 + 22 + C = 28 \implies 23 + C = 28 \implies C = 5 \]

### Step 6: Find A + C
Now we can find \(A + C\):
\[ A + C = 1 + 5 = 6 \]

### Final Answer
Thus, the value of \(A + C\) is:

\[ \boxed{6} \]