Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Write the rational expression in simplest form.\(\frac{2 x}{4 x+4}\)

Step-by-Step Solution

Answer

The rational expression in simplest form is: \( \frac{x}{2x+2} \)

Step 1: Factorize the denominator

To simplify the expression, first of all factorize the denominator, i.e., \(4x+4\). Both terms have a common factor of 4. Factoring simplifies the denominator to \(4(x+1)\)

Step 2: Simplify the expression

Now the rational expression becomes:\(\frac{2x}{4*(x+1)}\).Now, we simplify the fraction (if possible). In this case, we see that both the numerator and denominator have the number \(2\) in common. We can simplify this proportion by dividing both the numerator and the denominator by \(2\) to get: \( \frac{x}{2*(x+1)} \)

Step 3: Final Simplification

The expression can be simplified further by writing 2*(x+1) as 2x+2, so the rational expression becomes: \( \frac{x}{2x+2} \)

Key Concepts

Factoring

Factoring is a powerful technique in algebra that involves breaking down expressions into their simpler "building blocks" or factors. When you factor an expression, you're essentially writing it as a product of other expressions. These smaller expressions or numbers multiply together to recreate the original.
For the provided problem, we have the denominator as \(4x + 4\). If we look closely, we can notice that both terms, \(4x\) and \(4\), share a common multiplier.

• We recognize that both have the number 4 as a shared component.
• This means that we can "pull out" or factor this common value from the entire expression.
• When we factor \(4\) out from \(4x + 4\), we're left with \(4(x + 1)\).

Factoring makes it easier to see other common terms and allows further simplification, setting us up for the next steps.

Common Factor

Identifying a common factor is an essential part of simplifying expressions. A common factor is a number or term that divides exactly into all terms of a given expression.
In our rational expression \(\frac{2x}{4(x+1)}\), we can see that both the numerator and the composite term in the denominator share a common number.

• Both the numerator \(2x\) and the factor in the denominator \(4\) have 2 as a common factor.
• When two parts of a fraction have a common factor, it means you can divide both the numerator and the denominator by this shared factor.
• This simplification step is crucial as it reduces the fraction to its simpler form.

Reducing expressions by leveraging common factors helps achieve the goal of simplification more efficiently.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms, making them easier to work with or understand. It's important to note that while simplifying, the value of the expression doesn't change; it just becomes more streamlined.
With our expression \(\frac{x}{2(x+1)}\), we initially factor the terms as much as possible and look for any common factors, as previously discussed. Here's how it all gels together:

• From the numerically reduced form \(\frac{2x}{4(x+1)}\), we simplified it to \(\frac{x}{2(x+1)}\) by removing the common factor of 2.
• This process maintains the equivalency of the expression, ensuring its value stays unchanged.

The outcome is a rational expression in its simplest form. Simplifying not only serves an academic purpose but also enhances computational efficiency in complex mathematical problems.