Problem 92 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{10 z}{z+4}+z$$

Step-by-Step Solution

Answer

Combine the expressions as \( \frac{z^2 + 14z}{z+4} \).

Step 1: Identify the Rational Expressions

We have two terms: 1. \( \frac{10z}{z+4} \) 2. \( z \) Our goal is to combine these expressions by performing the given operations.

Step 2: Determine the Common Denominator

The first term already has a denominator of \( z+4 \). To perform the addition, we need to rewrite the second term \( z \) with the same denominator. Express \( z \) as \( \frac{z(z+4)}{z+4} = \frac{z^2 + 4z}{z+4} \).

Step 3: Combine the Terms

Now that both terms have the common denominator \( z+4 \), we can add them together:\[ \frac{10z}{z+4} + \frac{z^2 + 4z}{z+4} = \frac{10z + z^2 + 4z}{z+4}. \]

Step 4: Simplify the Expression

Combine like terms in the numerator:\[ \frac{z^2 + 14z}{z+4}. \] The numerator is now \( z^2 + 14z \), and the expression can't be simplified further since there are no common factors in the numerator and the denominator.

Key Concepts

Addition of Rational Expressions

Adding rational expressions is a vital skill that involves combining expressions with similar structures, like fractions, to form a new expression. With fractions, you must ensure you work with the same denominator; the same applies to rational expressions. In this case, we began with two terms:

• The rational expression \( \frac{10z}{z+4} \)
• The lone variable \( z \)

The task was to add these together. When adding rational expressions, remember that the key step is to first express both terms using the same denominator. This allows them to be added together smoothly, just like you would add fractions once their denominators match.

Common Denominator

Understanding the common denominator is crucial when adding rational expressions. A common denominator is like a shared footing that lets you unify different expressions for easier arithmetic, whether you're adding, subtracting, or comparing them.
For the expression \( \frac{10z}{z+4} + z \), we needed a common denominator so that the terms could be combined into a single expression. The first term, \( \frac{10z}{z+4} \), already has a denominator \( z+4 \). Our task was to express the loose term \( z \) with this same denominator:

• To equip \( z \) with a denominator \( z+4 \), we rewritten it as \( \frac{z(z+4)}{z+4} = \frac{z^2+4z}{z+4} \)

Once both terms shared the denominator \( z+4 \), we could proceed to add them.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form. This process is akin to reducing fractions by ensuring there are no common factors left in the numerator and denominator.
Once we had both terms over the common denominator \( z+4 \):

• We added them: \( \frac{10z}{z+4} + \frac{z^2 + 4z}{z+4} = \frac{10z + z^2 + 4z}{z+4} \)
• This resulted in: \( \frac{z^2 + 14z}{z+4} \)

The next task was to simplify. We combined like terms in the numerator to find it couldn’t be simplified further due to the lack of common factors between the numerator \( z^2 + 14z \) and the denominator \( z+4 \). This is key: always look for common factors to ensure expressions are in their simplest form.