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Chapter 7: Problem 29

For exercises \(5-48\), simplify. $$ \frac{z^{2}}{z+3}-\frac{5 z+24}{z+3} $$

Short Answer

Expert verified
\[ \frac{z^2 - 5z - 24}{z+3} \]

Step by step solution

01

Identify the common denominator

Both fractions have the same denominator, which is \(z+3\). This makes it easier to combine the fractions.
02

Combine the fractions

Since the denominators are the same, the fractions can be combined into a single fraction: \[ \frac{z^2}{z+3} - \frac{5z + 24}{z+3} = \frac{z^2 - (5z + 24)}{z+3} \]
03

Simplify the numerator

Distribute the negative sign and combine like terms: \[z^2 - 5z - 24\]}
04

Write the final simplified fraction

The final expression becomes: \[\frac{z^2 - 5z - 24}{z+3} \].

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Key Concepts

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

Common Denominator
When simplifying algebraic expressions involving fractions, a common denominator is crucial. The common denominator is the shared bottom part of the fractions, which allows you to combine them. In the provided exercise, both fractions \(\frac{z^2}{z+3}\) and \(\frac{5z + 24}{z+3}\) have the same denominator: \(z+3\). Since the denominators are identical, this makes our job easier because we can directly combine the fractions without further manipulation of the denominators. This helps simplify complex algebraic operations.
Combining Fractions
Combining fractions with a common denominator is straightforward. You keep the common denominator and subtract or add the numerators. In our exercise, the combined fraction is: \[ \frac{z^2}{z+3} - \frac{5z + 24}{z+3} = \frac{z^2 - (5z + 24)}{z+3} \]
By doing this, we have a single fraction with one simple denominator, which makes further simplification easier.
Distributing Negative Sign
When a negative sign is in front of a parenthesis, it needs to be distributed to all terms inside the parenthesis. Here鈥檚 how you do it:
In the numerators: \(z^2 - (5z + 24)\), the negative sign distributes to both \(5z\) and \(24\). This operation transforms the expression to: \[z^2 - 5z - 24 \]
Remember, distributing the negative sign changes the signs of the terms inside the parenthesis.
Combining Like Terms
Combining like terms simplifies an expression to its most reduced form. Like terms are terms that have the same variables raised to the same power. For example, \(z^2\) and \(z\) are unlike terms.
In the numerator from our exercise, we have \z^2 - 5z - 24\. The terms \(z^2, -5z,\) and \(-24\) are unlike terms, so there are no further simplifications needed in the numerator.
After combining fractions and distributing the negative sign, the final step leaves us with the simplified expression: \[ \frac{z^2 - 5z - 24}{z+3} \]

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Most popular questions from this chapter

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(14.9\) cases of syphilis per 100,000 Americans with a total of 45,834 cases of syphilis. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc.gov, 2011)

For exercises 43-58, (a) solve. (b) check. $$ \frac{4}{a+6}=\frac{9}{a-4} $$

For exercises 43-58, (a) solve. (b) check. $$ \frac{2}{x}=0 $$

For exercises \(67-82\), use the five steps and a proportion. About five of 100 pregnant women have pre-eclampsia, a condition that results in high blood pressure. About 300,000 pregnant women per year in the United States have pre-eclampsia. Find the number of pregnant women in the United States used to create this ratio. (Source: www.nytimes.com, March 17, 2009)

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{9} x+\frac{5}{3}=\frac{5}{9} x+\frac{7}{3} $$
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