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Chapter 8: Problem 239

In the following exercises, simplify. $$ \frac{-5 n-5}{n^{2}+n-6}+\frac{n+1}{2-n} $$

Short Answer

Expert verified
The simplified expression is \( \frac{-n^2 - 9n - 8}{(n + 3)(n - 2)} \).

Step by step solution

01

- Factor the Denominators

The first denominator is a quadratic expression. Factor it: \( n^2 + n - 6 = (n + 3)(n - 2) \). The second denominator \( 2 - n \) can be rewritten as \(-(n - 2)\).
02

- Rewrite the Rational Expressions

Rewrite the fractions using the factored forms of their denominators: \( \frac{-5n - 5}{(n + 3)(n - 2)} + \frac{n + 1}{-(n - 2)} \). Notice that \ \( -(n - 2) \) matches \( (n - 2) \) from the first denominator.
03

- Simplify the Second Fraction

Simplify the second fraction: \( \frac{n + 1}{-(n - 2)} = -\frac{n + 1}{n - 2} \). This makes the equation: \( \frac{-5n - 5}{(n + 3)(n - 2)} - \frac{n + 1}{n - 2} \).
04

- Combine the Fractions

Find a common denominator: \( (n + 3)(n - 2) \). Rewrite the second fraction: \( -\frac{(n + 1)(n + 3)}{(n + 3)(n - 2)} \). Now the combined expression is: \( \frac{-5n - 5 - (n + 1)(n + 3)}{(n + 3)(n - 2)} \).
05

- Simplify the Numerator

Expand and simplify the numerator: \( -5n - 5 - (n^2 + 4n + 3) = -n^2 - 9n - 8 \).
06

- Simplified Result

The simplified expression is: \( \frac{-n^2 - 9n - 8}{(n + 3)(n - 2)} \). This fraction cannot be simplified further.

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Key Concepts

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

Factoring Denominators
When dealing with rational expressions, it鈥檚 essential to factor the denominators if possible. Factoring helps break down the expression into simpler components, making it easier to work with.
Common Denominator
To combine rational expressions, we need a common denominator. This common denominator should be a multiple of the denominators of each fraction involved.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions involves several steps.

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