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Chapter 7: Problem 22

In the following exercises, simplify each rational expression. $$ \frac{5-d}{d-5} $$

Short Answer

Expert verified
-1

Step by step solution

01

Understand the Problem

Identify the rational expression that needs to be simplified, which is \( \frac{5-d}{d-5} \). Notice the relationship between the numerator and the denominator.
02

Factor the Numerator and Denominator

Observe that \(5-d\) can be rewritten as \(-1(d-5)\) because \(5-d = -1(d-5)\).
03

Simplify the Expression

Rewrite the rational expression using the factored form: \( \frac{5-d}{d-5} = \frac{-1(d-5)}{d-5} \). Since \(d-5\) appears in both the numerator and denominator, they cancel each other out, leaving \( -1 \).
04

Write the Final Answer

The simplified form of the expression is \( -1 \).

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Key Concepts

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

factoring
Factoring is a key skill needed to simplify rational expressions. It involves rewriting a polynomial as a product of its simplest components, called factors. In the exercise, we faced the expression \(5-d\). Notice that \(5-d\) can be factored by rewriting it in another form to see the relationship with \(d-5\).
Specifically, \(5-d\) can be expressed as \(-1(d-5)\). This is because if you distribute the \(-1\) back out, you'll get \(5 - d\) again.
To practice factoring, try to recognize patterns and use known identities, like how the expression \(a-b = -1(b-a)\). This understanding simplifies the process of working with rational expressions.
numerator and denominator relationship
Understanding the relationship between the numerator and the denominator is crucial. In the simplified expression \( \frac{5-d}{d-5} \), we notice that the numerator and denominator are not in their simplest forms yet.
By examining the terms closely, we recognize that the numerator \(5-d\) is the negative of the denominator \(d-5\). This relationship is important because it allows us to rewrite the fraction cleverly, making it easier to simplify.
Recognizing such relationships helps to recognize and efficiently work with fractions, making simplification steps straightforward.
cancellation in fractions
Cancellation is a powerful technique for simplifying rational expressions. It involves removing common factors that appear in both the numerator and the denominator. For instance:
  • Rewrite the given fraction \( \frac{5-d}{d-5} \) as \( \frac{-1(d-5)}{d-5} \) because \( 5-d = -1(d-5) \).
  • Once the fraction is in this form, the terms \( d-5 \) in the numerator and the denominator can cancel out each other.
After cancellation, the fraction simplifies to just \( -1 \). Always look out for terms that are exactly the same (or negatives of each other) in the numerator and the denominator, as this can lead to simplification by cancellation.

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