Resolve into Partial Fractions
If `(1)/(x^(3)(x+3))=(1)/(Ax) - (1)/(Bx^(2))+(1)/(Cx^(3))-(1)/(D(x+3))` then find the values of A, B, C and D.

Answer

To resolve the expression \(\frac{1}{x^3(x+3)}\) into partial fractions of the form \(\frac{1}{Ax} - \frac{1}{Bx^2} + \frac{1}{Cx^3} - \frac{1}{D(x+3)}\), we can follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ \frac{1}{x^3(x+3)} = \frac{1}{Ax} - \frac{1}{Bx^2} + \frac{1}{Cx^3} - \frac{1}{D(x+3)} \]

### Step 2: Find a common denominator
The common denominator on the right side is \(Ax \cdot Bx^2 \cdot Cx^3 \cdot D(x+3)\). We can express the right side as:
\[ \frac{B C D (x+3) - A C D x + A B D x^2 - A B C x^3}{A B C D x^3 (x+3)} \]

### Step 3: Set the numerators equal
Now, we equate the numerators:
\[ 1 = B C D (x+3) - A C D x + A B D x^2 - A B C x^3 \]

### Step 4: Expand the right side
Expanding the right side gives:
\[ 1 = B C D x + 3 B C D - A C D x + A B D x^2 - A B C x^3 \]

### Step 5: Collect like terms
Rearranging the terms, we have:
\[ 1 = (-A B C)x^3 + (A B D - A C D)x^2 + (B C D - A C D)x + 3 B C D \]

### Step 6: Compare coefficients
Since the left side is a constant (1), we can compare coefficients:
1. Coefficient of \(x^3\): \(-A B C = 0\)
2. Coefficient of \(x^2\): \(A B D - A C D = 0\)
3. Coefficient of \(x\): \(B C D - A C D = 0\)
4. Constant term: \(3 B C D = 1\)

### Step 7: Solve the equations
From the first equation \(-A B C = 0\), we have two cases:
- Case 1: \(A = 0\) (not possible since \(A\) cannot be zero)
- Case 2: \(B = 0\) or \(C = 0\) (not possible since \(B\) and \(C\) cannot be zero either)

Thus, we can assume \(C \neq 0\) and \(A = 0\).

From the second equation, substituting \(A = 0\):
\[ 0 - A C D = 0 \implies 0 = 0 \quad \text{(holds true)} \]

From the third equation:
\[ B C D - 0 = 0 \implies B C D = 0 \quad \text{(holds true)} \]

From the fourth equation:
\[ 3 B C D = 1 \implies B C D = \frac{1}{3} \]

### Step 8: Assign values
We can assign values to \(A\), \(B\), \(C\), and \(D\):
- Let \(C = 1\)
- Then \(B D = \frac{1}{3}\)

Assuming \(B = 1\) and \(D = \frac{1}{3}\):
- \(A = 0\)
- \(B = 1\)
- \(C = 1\)
- \(D = \frac{1}{3}\)

### Final Values
Thus, the values are:
- \(A = 0\)
- \(B = 1\)
- \(C = 1\)
- \(D = \frac{1}{3}\)