Resolve `(2x^(3)+3x^(2)+5x+7)/((x+1)^(5))` into partial fractions.

Answer

To resolve the expression \(\frac{2x^3 + 3x^2 + 5x + 7}{(x + 1)^5}\) into partial fractions, we will follow these steps:

### Step 1: Identify the form of the partial fractions
Since the denominator is \((x + 1)^5\), the partial fraction decomposition will take the form:
\[ \frac{A}{(x + 1)} + \frac{B}{(x + 1)^2} + \frac{C}{(x + 1)^3} + \frac{D}{(x + 1)^4} + \frac{E}{(x + 1)^5} \]
where \(A\), \(B\), \(C\), \(D\), and \(E\) are constants that we need to determine.

### Step 2: Set up the equation
We can write:
\[ \frac{2x^3 + 3x^2 + 5x + 7}{(x + 1)^5} = \frac{A}{(x + 1)} + \frac{B}{(x + 1)^2} + \frac{C}{(x + 1)^3} + \frac{D}{(x + 1)^4} + \frac{E}{(x + 1)^5} \]

Multiplying both sides by \((x + 1)^5\) to eliminate the denominator gives:
\[ 2x^3 + 3x^2 + 5x + 7 = A(x + 1)^4 + B(x + 1)^3 + C(x + 1)^2 + D(x + 1) + E \]

### Step 3: Expand the right-hand side
Now we will expand the right-hand side:
1. \(A(x + 1)^4 = A(x^4 + 4x^3 + 6x^2 + 4x + 1)\)
2. \(B(x + 1)^3 = B(x^3 + 3x^2 + 3x + 1)\)
3. \(C(x + 1)^2 = C(x^2 + 2x + 1)\)
4. \(D(x + 1) = D(x + 1)\)
5. \(E\)

Combining these, we get:
\[ Ax^4 + (4A + B)x^3 + (6A + 3B + C)x^2 + (4A + 3B + 2C + D)x + (A + B + C + D + E) \]

### Step 4: Equate coefficients
Now we equate the coefficients from both sides:
- For \(x^4\): \(A = 0\)
- For \(x^3\): \(4A + B = 2\)
- For \(x^2\): \(6A + 3B + C = 3\)
- For \(x^1\): \(4A + 3B + 2C + D = 5\)
- For the constant term: \(A + B + C + D + E = 7\)

### Step 5: Solve the system of equations
From \(A = 0\), we substitute into the other equations:
1. \(B = 2\)
2. \(3B + C = 3 \Rightarrow 3(2) + C = 3 \Rightarrow C = -3\)
3. \(3B + 2C + D = 5 \Rightarrow 3(2) + 2(-3) + D = 5 \Rightarrow D = 1\)
4. \(B + C + D + E = 7 \Rightarrow 2 - 3 + 1 + E = 7 \Rightarrow E = 7\)

### Step 6: Write the final partial fraction decomposition
Substituting back the values of \(A\), \(B\), \(C\), \(D\), and \(E\):
\[ \frac{2}{(x + 1)^2} - \frac{3}{(x + 1)^3} + \frac{1}{(x + 1)^4} + \frac{7}{(x + 1)^5} \]

Thus, the partial fraction decomposition is:
\[ \frac{2}{(x + 1)^2} - \frac{3}{(x + 1)^3} + \frac{1}{(x + 1)^4} + \frac{7}{(x + 1)^5} \]