The partial fractions of `(3x^(4)+x^(3)+8x^(2)+x+2)/(x(x^(2)+1)^2)` are …..

Answer

To find the partial fractions of the expression \(\frac{3x^4 + x^3 + 8x^2 + x + 2}{x(x^2 + 1)^2}\), we will follow these steps:

### Step 1: Set up the partial fraction decomposition
Given the denominator \(x(x^2 + 1)^2\), we can express the fraction as:
\[ \frac{3x^4 + x^3 + 8x^2 + x + 2}{x(x^2 + 1)^2} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2} \]
where \(A\), \(B\), \(C\), \(D\), and \(E\) are constants to be determined.

### Step 2: Multiply through by the common denominator
Multiply both sides by \(x(x^2 + 1)^2\) to eliminate the denominators:
\[ 3x^4 + x^3 + 8x^2 + x + 2 = A(x^2 + 1)^2 + (Bx + C)x(x^2 + 1) + (Dx + E)x \]

### Step 3: Expand the right-hand side
Now we will expand the right-hand side:
1. \(A(x^2 + 1)^2 = A(x^4 + 2x^2 + 1)\)
2. \((Bx + C)x(x^2 + 1) = (Bx + C)(x^3 + x) = Bx^4 + Cx^3 + Bx^2 + Cx\)
3. \((Dx + E)x = Dx^2 + Ex\)

Combining these, we have:
\[ 3x^4 + x^3 + 8x^2 + x + 2 = (A + B)x^4 + (C)x^3 + (2A + B + D)x^2 + (C + E)x + A \]

### Step 4: Compare coefficients
Now we compare the coefficients from both sides:
1. Coefficient of \(x^4\): \(A + B = 3\)
2. Coefficient of \(x^3\): \(C = 1\)
3. Coefficient of \(x^2\): \(2A + B + D = 8\)
4. Coefficient of \(x\): \(C + E = 1\)
5. Constant term: \(A = 2\)

### Step 5: Solve the equations
From the constant term, we have:
\[ A = 2 \]
Substituting \(A = 2\) into the first equation:
\[ 2 + B = 3 \implies B = 1 \]
Now substituting \(A\) and \(B\) into the third equation:
\[ 2(2) + 1 + D = 8 \implies 4 + 1 + D = 8 \implies D = 3 \]
Now substituting \(C = 1\) into the fourth equation:
\[ 1 + E = 1 \implies E = 0 \]

### Step 6: Write the partial fractions
Now we have:
- \(A = 2\)
- \(B = 1\)
- \(C = 1\)
- \(D = 3\)
- \(E = 0\)

Thus, the partial fraction decomposition is:
\[ \frac{2}{x} + \frac{x + 1}{x^2 + 1} + \frac{3x}{(x^2 + 1)^2} \]

### Final Answer
The partial fractions of \(\frac{3x^4 + x^3 + 8x^2 + x + 2}{x(x^2 + 1)^2}\) are:
\[ \frac{2}{x} + \frac{x + 1}{x^2 + 1} + \frac{3x}{(x^2 + 1)^2} \]