Let's simplify the given expression step by step:

The expression is:

\[
\frac{1}{1-c+c^2} + \frac{1}{1+c+c^2} - \frac{2c}{1-c+c^2}
\]

### Step 1: Combine the terms with a common denominator.
The common denominator for the first and third terms is \( (1-c+c^2) \), and for the second term, it is \( (1+c+c^2) \).

So, let's rewrite the expression as:

\[
\frac{1 - 2c}{1-c+c^2} + \frac{1}{1+c+c^2}
\]

### Step 2: Find the common denominator.
The common denominator between \( 1-c+c^2 \) and \( 1+c+c^2 \) is \( (1-c+c^2)(1+c+c^2) \).

So, the expression can be rewritten as:

\[
\frac{(1 - 2c)(1+c+c^2) + (1)(1-c+c^2)}{(1-c+c^2)(1+c+c^2)}
\]

### Step 3: Simplify the numerator.
Let's expand the numerator:

\[
(1 - 2c)(1+c+c^2) = 1 + c + c^2 - 2c - 2c^2 - 2c^3 = 1 - c - c^2 - 2c^3
\]

The second part of the numerator is:

\[
1 \times (1-c+c^2) = 1 - c + c^2
\]

Adding these together:

\[
(1 - c - c^2 - 2c^3) + (1 - c + c^2) = 2 - 2c - 2c^3
\]

### Step 4: Write the final simplified expression.
So the simplified expression is:

\[
\frac{2 - 2c - 2c^3}{(1-c+c^2)(1+c+c^2)}
\]

You can factor out the common factor of 2 from the numerator:

\[
\frac{2(1 - c - c^3)}{(1-c+c^2)(1+c+c^2)}
\]

This is the simplified form of the given expression.

Would you like more details or have any questions about this solution?

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Here are 5 related questions you might be interested in:

1. How do you factorize expressions involving cubes?
2. What is the geometric interpretation of the given expression?
3. How would the expression change if \(c\) was a complex number?
4. Can you solve similar expressions using partial fractions?
5. How does this expression behave as \(c\) approaches 1?

**Tip:** Always check if the numerator and denominator can be factored further to simplify complex algebraic expressions.

Learn how to simplify the complex algebraic expression involving fractions and cubes step-by-step. This problem involves partial fraction decomposition and algebraic manipulation. Suitable for high school students in Grades 10-12.

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