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

Answer

To find the partial fractions of the expression \(\frac{8}{(1-x)(1+x)^2(1+x^2)}\), 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 denominators suggest the following form:
\[ \frac{8}{(1-x)(1+x)^2(1+x^2)} = \frac{A}{1-x} + \frac{B}{1+x} + \frac{C}{(1+x)^2} + \frac{D}{1+x^2} \]
where \(A\), \(B\), \(C\), and \(D\) are constants that we need to determine.

### Step 2: Multiply through by the Denominator
To eliminate the fractions, multiply both sides by the denominator \((1-x)(1+x)^2(1+x^2)\):
\[ 8 = A(1+x)^2(1+x^2) + B(1-x)(1+x)(1+x^2) + C(1-x)(1+x^2) + D(1-x)(1+x)^2 \]

### Step 3: Expand the Right Side
Now, we need to expand the right-hand side:
1. Expand \(A(1+x)^2(1+x^2)\)
2. Expand \(B(1-x)(1+x)(1+x^2)\)
3. Expand \(C(1-x)(1+x^2)\)
4. Expand \(D(1-x)(1+x)^2\)

### Step 4: Combine Like Terms
Combine all the terms from the expansions and group them by powers of \(x\).

### Step 5: Set Up a System of Equations
Since the left side equals 8 (which can be written as \(8 + 0x + 0x^2 + 0x^3 + 0x^4\)), we can set up a system of equations by equating the coefficients of corresponding powers of \(x\) from both sides.

### Step 6: Solve the System of Equations
From the coefficients, we will derive equations for \(A\), \(B\), \(C\), and \(D\).

1. Coefficient of \(x^0\): \(A + B + C + D = 8\)
2. Coefficient of \(x^1\): \(B - A + C - 2D = 0\)
3. Coefficient of \(x^2\): \(2A + B + C = 0\)
4. Coefficient of \(x^3\): \(2B - A = 0\)
5. Coefficient of \(x^4\): \(A = 0\)

### Step 7: Substitute and Solve
From \(A = 0\), substitute into the other equations to find \(B\), \(C\), and \(D\).

1. From \(2B = 0\), we find \(B = 0\).
2. Substitute \(A\) and \(B\) into \(2A + B + C = 0\) gives \(C = 0\).
3. Substitute into \(A + B + C + D = 8\) gives \(D = 8\).

### Step 8: Write the Final Partial Fraction Decomposition
Now substituting back into the partial fractions, we have:
\[ \frac{8}{(1-x)(1+x)^2(1+x^2)} = \frac{0}{1-x} + \frac{0}{1+x} + \frac{0}{(1+x)^2} + \frac{8}{1+x^2} \]
Thus, the final answer is:
\[ \frac{8}{1+x^2} \]