(1+x)/(1-x) Simplify

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (1+x)/(1-x)

The expression (1+x)/(1-x) can be simplified using a few simple algebraic techniques. Here's how:

Understanding the Expression

The expression (1+x)/(1-x) represents a fraction where the numerator is (1+x) and the denominator is (1-x). To simplify it, we aim to eliminate any common factors between the numerator and denominator.

Simplifying the Expression

1. Factorization: We can't factor out any common factors directly from the numerator and denominator.

2. Manipulating the Expression: While we can't factor out common factors, we can try manipulating the expression. We can multiply both the numerator and denominator by (1+x). This is allowed because multiplying by 1 (in the form of (1+x)/(1+x)) doesn't change the value of the expression.

   (1+x)/(1-x) * (1+x)/(1+x) = (1+x)^2 / (1-x)(1+x)

3. Expanding and Simplifying: Now, expand the squares and use the difference of squares pattern to simplify the denominator.

   (1+x)^2 / (1-x)(1+x) = (1 + 2x + x^2) / (1 - x^2)

4. Result: The simplified expression is (1 + 2x + x^2) / (1 - x^2).

Important Notes

• This expression can't be further simplified unless we have additional information about the value of x.
• The original expression (1+x)/(1-x) is undefined when x=1, as the denominator becomes zero. This means the simplified expression is also undefined when x=1.

Conclusion

The expression (1+x)/(1-x) can be simplified to (1 + 2x + x^2) / (1 - x^2). Remember that the simplified expression is undefined when x=1.