Problem 22 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{2 n^{2}-8 n}{4-n}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-2n\).

Step 1: Factor the Numerator

First, we need to factor the expression in the numerator, which is \(2n^2 - 8n\). This can be factored by taking the greatest common factor (GCF) out, which is \(2n\). Thus, \(2n^2 - 8n = 2n(n - 4)\).

Step 2: Rewrite the Fraction

Now that we've factored the numerator, rewrite the fraction using the factored form: \(\frac{2n(n-4)}{4-n}\).

Step 3: Simplify Further by Recognizing Opposite Terms

We notice that \(n-4\) in the numerator is similar but opposite to \(4-n\) in the denominator. Since \(n-4 = -(4-n)\), replace \(n-4\) with \(-(4-n)\). Thus, the expression becomes \(\frac{2n(-(4-n))}{4-n}\).

Step 4: Cancel Common Factors

Now, divide the common factor \(4-n\) from both numerator and denominator. This simplifies to \(-2n\).

Step 5: Simplified Expression

The result after canceling \(4-n\) and simplifying is \(-2n\).

Key Concepts

Factoring

To simplify an algebraic fraction, one of the first steps is often factoring. When we talk about factoring in algebra, we refer to breaking down a more complex expression into smaller, easily manageable parts called factors.

Let's consider the expression given in the problem: \(2n^2 - 8n\). The goal is to express this polynomial in terms of its factors.

Here's how you can approach it:

• Look at each term in the polynomial separately, \(2n^2\) and \(-8n\).
• Identify the common factors shared by both terms. Here, \(2n\) is a factor common to both \(2n^2\) and \(-8n\).
• Factor out the greatest common factor (GCF), which is a critical step to simplify the entire expression.

By factoring \(2n\) from both terms, the expression \(2n^2 - 8n\) simplifies to \(2n(n - 4)\). Now that it's factored, it makes simplifying the entire fraction easier.

Factoring is essential because it reduces complex expressions to manageable parts, preparing them for further simplification or solving.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. That means there are no like terms to combine, no common factors to cancel, and everything is as straightforward as possible.

In the context of our exercise, after factoring the numerator, we had the expression \(\frac{2n(n-4)}{4-n}\). To simplify this further, we need to look for common terms or factors in both the numerator and the denominator.

Here, we noticed that \(n-4\) is very similar to \(4-n\), just they are opposites. Hence:

• Recognize that \(n-4 = -(4-n)\), which allows us to express \(n-4\) as \(-(4-n)\). By doing this, the fraction then becomes \(\frac{2n(-(4-n))}{4-n}\).
• This step is crucial as having the same factor \(4-n\) in both numerator and denominator allows canceling.

Canceling common factors is what we do when simplifying because it removes redundant parts of the expression. Finally, once this cancellation is complete, the expression simplifies neatly to \(-2n\).

Simplifying expressions in this manner not only makes them easier to work with but also often unveils results that are more understandable and applicable in solving problems.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a vital tool in algebra for factoring expressions. The GCF is the largest factor shared between the terms in a polynomial, and identifying it correctly is the key to breaking down expressions.

In our exercise, the expression \(2n^2 - 8n\) required factoring. To identify the GCF, we followed these steps:

• Look at each term in the polynomial: \(2n^2\) and \(-8n\).
• Identify the prime factors and the common elements in both terms. Here, \(2n\) stands out.
• The GCF, \(2n\), is then factored out to get \(2n(n-4)\).

By factoring out the GCF, you effectively simplify the problem, making further steps more straightforward. The GCF helps streamline algebraic manipulation, allowing students to focus on more advanced concepts.

Once you learn to quickly spot and use the GCF, many other algebraic processes will become much more manageable, leading to concise and clear solutions.