\( 1.3 .2 \quad \frac { x ^ { 3 } + x ^ { 2 } + x } { x ^ { 3 } - 1 } + \frac { 2 x + 2 } { 1 - x ^ { 2 } } + \frac { x + 1 } { x ^ { 2 } - 1 } \)

To simplify the expression \( \frac { x ^ { 3 } + x ^ { 2 } + x } { x ^ { 3 } - 1 } + \frac { 2 x + 2 } { 1 - x ^ { 2 } } + \frac { x + 1 } { x ^ { 2 } - 1 } \), let's break it down step by step. First, we know: 1. The denominator \( x^3 - 1 \) can be factored as \( (x - 1)(x^2 + x + 1) \). 2. The principal denominator \( 1 - x^2 \) is equivalent to \( (1 - x)(1 + x) \), and it appears we need to express all terms with a common denominator. Revising the expression: - The first term is \( \frac{x^3 + x^2 + x}{(x-1)(x^2 + x + 1)} \). - The second term modifies to \( \frac{2(x + 1)}{-(x - 1)(x + 1)} = \frac{-2}{(x - 1)(1 + x)} \). - The third term can be written as \( \frac{x + 1}{(x - 1)(x + 1)} = \frac{1}{x - 1} \) (cancelling \( x + 1 \)). Combining all terms: You may rewrite it in terms of a common denominator of \( (x - 1)(x^2 + x + 1) \) or others as necessary and simplify accordingly. This involves finding the appropriate coefficients for each term and performing the algebra, which leads to combining like terms. However, the expansion and simplification of the rational functions may yield cancellations and lead to a clearer expression. In summary, to evaluate and simplify complex rational expressions: 1. Factor denominators. 2. Identify a common denominator. 3. Combine fractions and simplify. If you'd like a further breakdown of mistakes or more examples of real-world applications, let me know!