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

Options

A. `A+B+C=28`
B. `A+B-C=18`
C. `A-B= -21`
D. `2A+B=24`

Answer

To solve the equation

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

we will first express the right-hand side with a common denominator, which is \((x^2 + 1)^3\).

### Step 1: Combine the Right-Hand Side
We can rewrite the right-hand side as:

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

### Step 2: Expand the Numerator
Now we expand the numerator:

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

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

Combining these, the numerator becomes:

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

### Step 3: Set Numerators Equal
Now we set the numerators equal to each other:

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

### Step 4: Compare Coefficients
We can compare the 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: Substitute \(A\) into Other Equations
From \(A = 1\):

1. Substitute into the second equation:
\[ 2(1) + B = 24 \implies 2 + B = 24 \implies B = 22 \]

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

### Final Values
Thus, we have:

\[ A = 1, \quad B = 22, \quad C = 5 \]

### Summary
The values of \(A\), \(B\), and \(C\) are:

\[ A = 1, \quad B = 22, \quad C = 5 \]