Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Rational Expressions Simplify the expression. $$\frac{2}{z+2}-\left(3-\frac{2}{z}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-3z^2-2z+4}{(z+2)z} \)

Step 1: Rewrite the Problem

First, separate out the subtracted expression for better clarity. Rewrite it as: \[ \frac{2}{z+2}-3+\frac{2}{z} \]

Step 2: Find the Common Denominator

Find a common denominator to combine the fractions. Here it is \((z+2)z\). So, the expression would be rewritten as: \[ \frac{2z}{(z+2)z}-\frac{3(z+2)z}{(z+2)z}+\frac{2(z+2)}{(z+2)z} \]

Step 3: Simplify Numerator

In this step, multiply and simplify the numerators which gives: \[ \frac{2z-3z^2-6z+2z+4}{(z+2)z} \]

Step 4: Combine Like Terms

Combine like terms in the numerator: \[ \frac{-3z^2-2z+4}{(z+2)z} \]

Key Concepts

Working with Common Denominators

When dealing with rational expressions, finding a common denominator is essential for adding or subtracting fractions. A common denominator is a shared multiple of the denominators involved in an operation. It allows you to rewrite fractions so that they can be combined. In our example, the original expression involves the fractions \( \frac{2}{z+2} \) and \( \frac{2}{z} \).

To find a common denominator, consider the different terms in each denominator. For \( \frac{2}{z+2} \) and \( \frac{2}{z} \), the common denominator is the product of the two distinct terms: \((z+2)z\).

• This ensures that each fraction is expressed with the same denominator.
• Multipliers to reach the common denominator are applied to both the numerator and denominator.

By obtaining this shared base, you can combine the fractions into a single expression. This step simplifies further mathematical operations and makes the process of simplification and rearranging terms possible.

Simplifying Expressions

Simplifying expressions is a core skill in algebra that involves rewriting an expression in its simplest form. After finding a common denominator, the next step is to simplify the entire expression.

This involves:

• Multiplying out the terms in the numerators.
• Combining them into a single fraction.

In our problem, we expanded each numerator after adjusting for the common denominator:\( \frac{2z-3z^2-6z+2(z+2)}{(z+2)z} \).

It's important to carefully carry out multiplication to avoid mistakes. After expanding, collect all the terms over the common denominator into one single fraction. This simplified form prepares the expression for combining like terms to further reduce complexity.

Combining Like Terms

Once you have a single fraction, combining like terms is the final step in simplifying an expression. Like terms refer to terms in a polynomial that have the same variables raised to the same power.

In our problem, the expression \( \frac{2z-3z^2-6z+2z+4}{(z+2)z} \) includes terms that can be combined:

• Terms with \( z^2 \), \(-3z^2\).
• Linear terms with \( z \), these are \( 2z - 6z + 2z \).
• Constant terms like \( 4 \), which do not have variables.

The final simplified form becomes \( \frac{-3z^2-2z+4}{(z+2)z} \). Combining like terms reduces clutter and helps in achieving a more concise form of the expression.