\frac{2}{1 + a} + \frac{1}{a - 1} + \frac{3a}{1 - a^2} + \frac{a}{1 + a^3}
Answer (2)
To simplify the given expression 1 + a 2 + a − 1 1 + 1 − a 2 3 a + 1 + a 3 a , let's break it down step-by-step.
Identify Individual Fractions:
1 + a 2
a − 1 1
1 − a 2 3 a
1 + a 3 a
Recognize Common Patterns and Simplifications:
1 − a 2 = ( 1 + a ) ( 1 − a ) , so 1 − a 2 3 a can be rewritten as ( 1 + a ) ( 1 − a ) 3 a .
The cubic expression 1 + a 3 can be factored further but we'll consider it depending on the requirement for simplification.
Combine the Fractions:
Finding a common denominator may require further information about the relationships between the terms.
Unfortunately, without additional constraints or specific values of a , further simplification would involve finding a common denominator or specific substitutions based on any identities or given values. If the context allows, more algebraic manipulation can be applied to seek a simplified expression or specific evaluation.
In high school mathematics, understanding this process emphasizes recognizing factor patterns and potential simplifications that often show up in rational expressions.
To simplify the expression 1 + a 2 + a − 1 1 + 1 − a 2 3 a + 1 + a 3 a , factor differences of squares and identify common factors. Combine the fractions using a common denominator for simplification. Understanding these steps can greatly help in solving complex expressions in algebra.
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