Problem 38 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{4 x+8}{2 x} \cdot \frac{x^{2}}{x+2} $$

Step-by-Step Solution

Answer

The simplified expression is \(2x\).

Step 1: Factoring the Numerators and Denominators

First, let's factor the numerators and denominators of the fractions where possible. The expression can be rewritten as:\[\frac{4(x+2)}{2x} \cdot \frac{x^2}{x+2}\]Here, the numerator \(4x+8\) is factored into \(4(x+2)\).

Step 2: Cancelling Common Factors

Next, identify common factors in the numerators and denominators that can be cancelled. Notice that \(x+2\) is both in the numerator of the first fraction and the denominator of the second fraction, allowing us to cancel them out:\[\frac{4(x+2)}{2x} \cdot \frac{x^2}{x+2} = \frac{4}{2x} \cdot x^2\]

Step 3: Simplifying the Expression Further

Simplify the remaining expression. Multiply the numerators and the denominators:\[\frac{4x^2}{2x}\]Now, we can simplify the fraction by cancelling common factors. Divide the numerator and the denominator by 2:\[\frac{2x^2}{x}\]

Step 4: Final Simplification

Divide the variables in the simplified fraction. The expression simplifies to:\[2x\]

Step 5: Conclusion

The simplified form of the original expression is \(2x\). This is derived after factoring, cancelling common terms, and simplifying the resulting expression.

Key Concepts

Factoring

Factoring is all about breaking down an expression into simpler components called "factors." These factors, when multiplied together, recreate the original expression.
For instance, with the expression \(4x+8\), we can identify that both 4 and 8 share a common factor of 4.
So, we factor \(4x+8\) into \(4(x+2)\). Factoring is an essential step because it helps reveal potential opportunities to simplify expressions.

Think of factoring like unpacking a box to see what's inside. Once you see the individual components, it's easier to work with them.
This process is valuable, especially when simplifying algebraic fractions, because it often reveals ways to cancel out unnecessary parts.
Remember, always check if there is a greatest common factor that can be pulled out.

• Identify common factors in expressions.
• Break down complex expressions into multipliers.
• Make simplification steps easier through factoring.

By mastering factoring, you make the entire process of working with algebraic expressions more straightforward and less intimidating.

Canceling Common Factors

Once you've factored out expressions, the next logical step is to look for common factors that can be cancelled.
Cancelling is the practice of removing identical factors from the numerator and denominator of a fraction.
For example, in the expression \(\frac{4(x+2)}{2x} \cdot \frac{x^2}{x+2}\), the common factor \(x+2\) appears in both a numerator and a denominator.

This allows us to remove the \(x+2\) from each, simplifying our expression.
It's a bit like simplifying a recipe by removing unnecessary ingredients that cancel each other out. When factors resemble each other across the division line, cancel them to streamline the fraction.

• Reduce the complexity of expressions.
• Make sure to equally balance any removed factors.
• Know which terms to eliminate for simplifying equations.

This step is crucial because it leads to a more manageable expression that is easier to solve or further simplify.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions.
Dealing with these requires understanding how to handle variables alongside numbers.
In our original expression \(\frac{4x+8}{2x} \cdot \frac{x^2}{x+2}\), each fraction has numerators and denominators with variables.

The steps to simplifying algebraic fractions involve factoring, cancelling, and rewriting the expression for a clearer representation.
This makes an initially complex fraction easier to interpret and work with. Like regular fractions, the core operations for manipulating algebraic fractions include multiplying the numerators and dividing by the product of denominators.

• View fractions as division of expressions.
• Apply standard fraction operations to expressions.
• Simplify by reducing common terms.

Understanding algebraic fractions ensures that you can handle and simplify equations efficiently, paving the way for solving more complex problems.