Resolve into Partial Fractions
(vii) `(1)/(x^(3)(x+a))`

Answer

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

### Step 1: Set Up the Partial Fraction Decomposition
We start by expressing the given fraction in terms of its partial fractions. The expression can be decomposed as follows:

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

### Step 2: Combine the Right-Hand Side
Next, we will combine the right-hand side into a single fraction. The common denominator will be \( x^3(x+a) \):

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

### Step 3: Set the Numerators Equal
Now, we equate the numerators from both sides:

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

### Step 4: Expand the Right-Hand Side
Expanding the right-hand side gives us:

\[ 1 = A x^3 + A a x^2 + B x^2 + B a x + C x + C a + D x^3 \]

Combining like terms results in:

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

### Step 5: Equate Coefficients
Since the left-hand side is a constant (1), we can equate the coefficients of corresponding powers of \( x \):

1. Coefficient of \( x^3 \): \( A + D = 0 \)
2. Coefficient of \( x^2 \): \( A a + B = 0 \)
3. Coefficient of \( x^1 \): \( B a + C = 0 \)
4. Constant term: \( C a = 1 \)

### Step 6: Solve the System of Equations
From the equations, we can solve for \( A, B, C, \) and \( D \):

1. From \( C a = 1 \), we get \( C = \frac{1}{a} \).
2. Substitute \( C \) into \( B a + C = 0 \):
\[ B a + \frac{1}{a} = 0 \implies B a = -\frac{1}{a} \implies B = -\frac{1}{a^2} \]
3. Substitute \( B \) into \( A a + B = 0 \):
\[ A a - \frac{1}{a^2} = 0 \implies A a = \frac{1}{a^2} \implies A = \frac{1}{a^3} \]
4. Substitute \( A \) into \( A + D = 0 \):
\[ \frac{1}{a^3} + D = 0 \implies D = -\frac{1}{a^3} \]

### Step 7: Write the Final Partial Fraction Decomposition
Now we can substitute the values of \( A, B, C, \) and \( D \) back into the partial fractions:

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

### Final Answer
Thus, the final answer for the partial fraction decomposition is:

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

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