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Chapter 6: Problem 35

Write each rational expression in lowest terms. $$ \frac{(x+1)(x-1)}{(x+1)^{2}} $$

Short Answer

Expert verified
\( \frac{x-1}{x+1} \)

Step by step solution

01

- Factor the numerator and the denominator

The given rational expression is \( \frac{(x+1)(x-1)}{(x+1)^{2}} \). The numerator is already factored as \((x+1)(x-1)\), and the denominator is \((x+1)^{2}\).
02

- Identify common factors

Both the numerator \((x+1)(x-1)\) and the denominator \((x+1)^{2}\) have a common factor of \((x+1)\).
03

- Cancel the common factors

Cancel the common factor \((x+1)\) from the numerator and the denominator: \[ \frac{(x+1)(x-1)}{(x+1)^{2}} = \frac{(x-1)}{(x+1)} \]
04

- Write the simplified expression

After canceling the common factor, the simplified rational expression is \( \frac{x-1}{x+1} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is the first step in simplifying many rational expressions. It involves breaking down a polynomial into a product of its simplest parts, known as factors. For instance, the polynomial \(x^2 - 1\) can be factored into \((x+1)(x-1)\). Each factor represents a simpler expression that, when multiplied together, gives back the original polynomial. Factoring is crucial because it allows us to see the underlying structure of the polynomial, making it easier to identify common factors with the denominator in a rational expression.
Common Factors
Common factors are elements that appear in both the numerator and the denominator of a rational expression. In the given exercise, \( (x+1) \) is a common factor in both the numerator \((x+1)(x-1)\) and the denominator \((x+1)^2\). Recognizing common factors is essential because it allows us to simplify the expression. By identifying and eventually canceling out these common factors, we make the rational expression simpler and easier to work with.
Canceling Factors
Canceling factors involves removing the common factors from both the numerator and the denominator of a rational expression. This process simplifies the expression. For example, in the expression \( \frac{(x+1)(x-1)}{(x+1)^2} \), the factor \( (x+1) \) appears in both the numerator and the denominator. By canceling \( (x+1) \) from both, the expression simplifies to \( \frac{x-1}{x+1} \). Always ensure that you only cancel factors, not terms, as this can otherwise make the expression incorrect.
Rational Expressions
Rational expressions are fractions that have polynomials in the numerator and the denominator. They can often be simplified by factoring and canceling common factors. The goal is to express the rational expression in its simplest form, where no further factors can be canceled. For instance, \( \frac{(x+1)(x-1)}{(x+1)^2} \) simplifies to \( \frac{x-1}{x+1} \) by factoring and canceling common factors. Simplifying rational expressions is a fundamental skill in algebra, helping to solve equations and understand functions.

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