Resolve into partial fractions `(x^(4)+3x+1)/(x^(3)(x+1))`.

Answer

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

### Step 1: Set up the partial fraction decomposition
We start by expressing the given rational function as a sum of simpler fractions. Since the denominator is \(x^3(x + 1)\), we can write:

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

where \(A\), \(B\), \(C\), and \(D\) are constants we need to determine.

### Step 2: Combine the right-hand side
Next, we will combine the right-hand side over a common denominator:

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

### Step 3: Expand the numerator
Now, we will expand the numerator:

\[ A(x^2)(x + 1) = Ax^3 + Ax^2 \]
\[ B(x)(x + 1) = Bx^2 + Bx \]
\[ C(x + 1) = Cx + C \]
\[ D(x^3) = Dx^3 \]

Combining these, we have:

\[ (A + D)x^3 + (A + B)x^2 + (B + C)x + C \]

### Step 4: Set up the equation
Now we set the numerator equal to the numerator of the original expression:

\[ (A + D)x^3 + (A + B)x^2 + (B + C)x + C = x^4 + 0x^3 + 0x^2 + 3x + 1 \]

### Step 5: Compare coefficients
From the equation above, we can compare coefficients:

1. For \(x^3\): \(A + D = 0\)
2. For \(x^2\): \(A + B = 0\)
3. For \(x\): \(B + C = 3\)
4. For the constant term: \(C = 1\)

### Step 6: Solve the system of equations
From \(C = 1\), we substitute into the third equation:

\[ B + 1 = 3 \implies B = 2 \]

Now substitute \(B = 2\) into the second equation:

\[ A + 2 = 0 \implies A = -2 \]

Finally, substitute \(A = -2\) into the first equation:

\[ -2 + D = 0 \implies D = 2 \]

### Step 7: Write the final partial fraction decomposition
Now we have:

- \(A = -2\)
- \(B = 2\)
- \(C = 1\)
- \(D = 2\)

Thus, we can write the partial fraction decomposition as:

\[ \frac{x^4 + 3x + 1}{x^3(x + 1)} = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{x^3} + \frac{2}{x + 1} \]

### Final Answer
The resolved partial fractions are:

\[ \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{x^3} + \frac{2}{x + 1} \]

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