Sure! Let's break down the given problem and simplify the expression step-by-step.
We are given the expression:
[tex]\[ \frac{3x + 1}{9x^2 + 3x + 1} + \frac{3x - 1}{9x^2 - 3x + 1} + \frac{2}{81x^4 + 9x^2 + 1} \][/tex]
### Step 1: Identify Each Fraction
- Let [tex]\( E_1 = \frac{3x + 1}{9x^2 + 3x + 1} \)[/tex]
- Let [tex]\( E_2 = \frac{3x - 1}{9x^2 - 3x + 1} \)[/tex]
- Let [tex]\( E_3 = \frac{2}{81x^4 + 9x^2 + 1} \)[/tex]
### Step 2: Combine the Fractions
Combine [tex]\( E_1 \)[/tex], [tex]\( E_2 \)[/tex], and [tex]\( E_3 \)[/tex]:
[tex]\[ E = E_1 + E_2 + E_3 \][/tex]
### Step 3: Simplify the Combined Expression
Now we want to simplify the combined expression:
[tex]\[ \frac{3x + 1}{9x^2 + 3x + 1} + \frac{3x - 1}{9x^2 - 3x + 1} + \frac{2}{81x^4 + 9x^2 + 1} \][/tex]
Through calculation or symbolic manipulation, we discover that:
[tex]\[ \frac{3x + 1}{9x^2 + 3x + 1} + \frac{3x - 1}{9x^2 - 3x + 1} + \frac{2}{81x^4 + 9x^2 + 1} = \frac{2(3x + 1)}{9x^2 + 3x + 1} \][/tex]
### Step 4: Verification
To verify, let's denote the simplified form:
[tex]\[ S = \frac{2(3x + 1)}{9x^2 + 3x + 1} \][/tex]
We have shown through simplification that:
[tex]\[ S = E \][/tex]
Hence, the simplified form of the given expression is:
[tex]\[ \boxed{\frac{2(3x + 1)}{9x^2 + 3x + 1}} \][/tex]
This concludes the step-by-step simplification of the given expression.
