Problem 7 - Elementary and Intermediate Algebra

Question

The sum of two rational expressions is \(\frac{4 x+4}{5(x+1)} .\) Factor the numerator and then simplify the result.

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4}{5} \).

Step 1: Identify the numerator

The given rational expression has the numerator as \(4x + 4\).

Step 2: Factor out the common term

Notice that both terms in the numerator \(4x + 4\) have a common factor of 4. Thus, we can factor out 4, resulting in \(4(x + 1)\).

Step 3: Rewrite the expression with the factored numerator

Substitute the factored numerator back into the original rational expression, which gives \( \frac{4(x + 1)}{5(x + 1)} \).

Step 4: Simplify the expression

Since \(x + 1\) is present in both the numerator and the denominator, they can be cancelled out, simplifying the expression to \( \frac{4}{5} \).

Key Concepts

Factoring

Factoring is an essential tool used in algebra to simplify expressions and solve equations. When you factor an expression, you look for common factors in the terms that make up the expression, breaking it down into simpler forms for easier manipulation. In our exercise, we started with the numerator, which was a polynomial: \(4x + 4\). By identifying common factors, we discovered that both terms in the polynomial shared the number 4 as a factor. Factoring out the common factor of 4, we rewrote the numerator as \(4(x + 1)\). This is an example of a linear polynomial, and recognizing this was the first step in simplifying the rational expression. Factoring simplifies the process of cancellation later, especially when similar terms appear in both the numerator and denominator.

Simplifying Expressions

Simplifying expressions is all about making math problems easier to deal with by reducing them to their simplest form. After factoring the numerator in our rational expression, the next step was to cancel out common terms in both the numerator and the denominator. In the exercise, once we had factored the numerator to \(4(x + 1)\), we observed that the \(x + 1\) term was also present in the denominator. Because these terms appeared unchanged in both the numerator and the denominator, they could be cancelled out entirely. Applying this cancellation rule is a key step in simplifying rational expressions.After the cancellation, we were left with a very simple expression: \(\frac{4}{5}\). Simplifying not only makes computation more manageable, but it also helps you identify the true form of the expression that might not be visible at first glance.

Numerator and Denominator

In a fraction, the numerator is the top part, while the denominator is at the bottom. These two components are crucial in understanding and manipulating rational expressions. For any rational expression, like \(\frac{4(x + 1)}{5(x + 1)}\), dealing with the numerator and denominator properly can lead to effective simplification. The numerator and denominator often have interconnected parts that can be factored or simplified. In our example, once we factored the numerator to \(4(x + 1)\), it was clear that both \(x + 1\) terms could be cancelled due to their presence in both parts of the fraction. This highlights the importance of observing both sections of a rational expression closely.Knowing how to handle the numerator and the denominator is key to working through problems efficiently and reaching an answer that is both correct and easy to interpret.