Problem 29 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the common denominator

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

Step 2: Combine the fractions

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

Step 3: Simplify the numerator

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

Step 4: Write the final simplified fraction

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

Key Concepts

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’s 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: \[ \frac{z^2 - 5z - 24}{z+3} \]