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Chapter 6: Problem 32

Write each rational expression in lowest terms. $$ \frac{(2 x+7)(x-1)}{(2 x+3)(2 x+7)} $$

Short Answer

Expert verified
\frac{x - 1}{2x + 3}

Step by step solution

01

Identify Common Factors

First, identify any common factors in the numerator and the denominator. In this case, both the numerator \( (2x + 7)(x - 1) \) and the denominator \( (2x + 3)(2x + 7) \) share the common factor \( 2x + 7 \).
02

Cancel Out Common Factors

Next, cancel out the common factor \( 2x + 7 \) from both the numerator and the denominator. This leaves us with: \[\frac{x - 1}{2x + 3} \]
03

Write the Simplified Expression

Finally, write the simplified rational expression. After the common factor is canceled, the simplified expression is \[\frac{x - 1}{2x + 3} \].

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Key Concepts

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

common factors in algebra
In algebra, common factors are terms that can be found in both the numerator and the denominator of a rational expression. Identifying these common factors is the first key step in simplifying rational expressions.
For example, consider the rational expression \( \frac{(2 x+7)(x-1)}{(2 x+3)(2 x+7)} \). Here, \( 2x + 7 \) is a common factor that appears in both the numerator and the denominator.
Understanding how to find these common factors is essential. Start by factoring the numerator and the denominator fully. Look for terms that are identical in both to identify the common factors.
canceling common factors
Once we've identified common factors, the next step is to cancel them out. Canceling common factors simplifies the rational expression and makes it easier to work with.
For our example, in \( \frac{(2 x+7)(x-1)}{(2 x+3)(2 x+7)} \), we can see that \( 2x + 7 \) is present in both the numerator and the denominator. By canceling this common factor, we are left with \( \frac{x - 1}{2x + 3} \).
When canceling out the common factors, make sure to only cancel factors that are multiplied, not added or subtracted. This means we only cancel terms that are products, not sums or differences.
rational expressions simplification
Simplifying rational expressions is the process of reducing them to their simplest form. This involves the steps of identifying and canceling common factors.
Let's consider the expression \( \frac{(2 x+7)(x-1)}{(2 x+3)(2 x+7)} \).
  • First, factorize both the numerator and denominator if needed. In this case, they are already factored.
  • Next, identify any common factors, which here is \( 2x + 7 \).
  • Then, cancel out the common factors to simplify the expression to \( \frac{x - 1}{2x + 3} \).
By following these steps, you can simplify most rational expressions, making them easier to understand and work with in algebraic operations and equations.

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Most popular questions from this chapter

A student was asked to simplify the rational expression $$ \frac{2 x+y}{4 x+2 y} $$ and wrote the incorrect answer \(\frac{x+y}{4 x+y}\) WHAT WENT WRONG? What is the correct answer?

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