Problem 10 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$ \frac{18}{2 x+4} $$

Step-by-Step Solution

Answer

The simplest form of \(\frac{18}{2x+4}\) is \(\frac{9}{x+2}.\)

Step 1: Factor the denominator

In the given expression \(\frac{18}{2x+4}\), factor the denominator to get \(\frac{18}{2(x+2)}\). Factoring is important because it simplifies the equation and potentially allows for cancellation of like terms.

Step 2: Simplify the expression

Divide the numerator and denominator by the common factor 2, resulting in \(\frac{9}{(x+2)}\). This is the simplified form of \(\frac{18}{2x+4}\)

Step 3: Confirm if further simplification is possible

Check if there are more common factors in the numerator and the denominator, or if further factoring is possible. In this case, there are none, so no further simplification is possible.

Key Concepts

Factoring Expressions

Factoring is a key mathematical skill that simplifies expressions and reveals hidden relationships. When we talk about factoring, we're breaking down an expression into simpler components called factors. This helps in simplifying or solving equations swiftly. Think of it as identifying ingredients in a recipe.

For instance, if we look at the denominator of our example expression, \(2x+4\), we can factor it by finding a common factor. Both terms, \(2x\) and 4, share a common factor, which is 2. By factoring out the 2, we rewrite the expression as \(2(x+2)\).

This simplified form is more manageable and enables further steps of simplification. Recognizing common factors quickly is essential in algebra to make expressions easier to manage.

Simplifying Algebraic Fractions

Simplifying algebraic fractions means reducing them to their simplest form. This process is akin to simplifying a regular fraction by removing common factors in the numerator and denominator.

In our original exercise, we had the expression \(\frac{18}{2x+4}\). Once we've factored the denominator to \(2(x+2)\), the next step is to simplify the fraction by canceling out common factors. We notice that the number 18 in the numerator and 2 in \(2(x+2)\) as a part of the denominator have a greatest common factor of 2.

By dividing both the numerator and the denominator by 2, we get \(\frac{9}{x+2}\). This simplification makes the expression easier to understand and use in further calculations or applications. Recognizing common factors is essential to this process.

Numerator and Denominator

The terms numerator and denominator are fundamental components of any fraction or rational expression. The numerator is the top part of the fraction, representing how many parts of the whole are being considered. The denominator, on the other hand, is the bottom part and it tells us how many parts make up a whole.

In our exercise, 18 is the numerator, and \(2x+4\) is initially the denominator. Understanding these terms helps in executing operations such as addition, subtraction, multiplication, and division with fractions.

In algebraic fractions, it's useful to manage and simplify these components to get an expression in its simplest form. Simplification is primarily about breaking down the numerator and denominator to their essence, removing unnecessary complexity, like shared factors, to make the fraction more efficient in calculations.