Problem 45 - Elementary and Intermediate Algebra

Question

Simplify. See Example 4. $$ \frac{x+3}{3 x+9} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{3}\), for \(x \neq -3\).

Step 1: Factor the Numerator

Let's factor the numerator, which is \(x + 3\). Since there are no common factors other than 1, the numerator is already simplified as \(x + 3\).

Step 2: Factor the Denominator

Now, let's factor the denominator \(3x + 9\). Notice that both terms have a common factor of 3, so we can factor out a 3: \[3x + 9 = 3(x + 3)\].

Step 3: Simplify the Expression

The expression now looks like this: \[\frac{x + 3}{3(x + 3)}\]. We can cancel the \(x + 3\) term in the numerator and the denominator because they are common factors and do not equal zero. This leaves us with: \[\frac{1}{3}\].

Step 4: Confirm Domain Restrictions

Before cancelling, ensure that \(x+3eq0\) to avoid division by zero. Thus, \(x eq -3\). The simplified expression for all permissible \(x\) is \(\frac{1}{3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a critical step in handling rational expressions. It involves reducing the complexity by getting rid of unnecessary components, much like cleaning up to make things clearer.

When we simplify fractions, our goal is to express the fraction in its simplest form. This means removing any common factors from the numerator and the denominator.

Taking a closer look at our exercise, we notice that the expression is \( \frac{x+3}{3x+9} \). To simplify this, we first consider whether the numerator and the denominator have common factors.

In this case, the numerator already appears in its simplest form since there are no factors other than 1. However, the denominator \( 3x + 9 \) can be factored. Once both the numerator and denominator are fully simplified, any identical factors can be canceled out.

This process leaves us with a simpler fraction. In our example, after proper simplification by cancelling common factors, we get \( \frac{1}{3} \).

• Simplify by identifying and removing common factors.
• A simpler expression is easier to work with and understand.
• Verify that simplification doesn’t lead to errors or omissions.

Factoring Polynomials

Factoring polynomials is a crucial skill needed for simplifying rational expressions. It's all about breaking down expressions into their simplest multipliers, or factors.

In our exercise, the denominator \( 3x + 9 \) can be factored by noticing that each term shares a common factor. In this instance, both terms, \(3x\) and \(9\), can be divided by 3. So, by factoring out 3, we get:\[ 3x + 9 = 3(x + 3) \]

Why Factor?

Factoring helps:

• Identify common factors between terms.
• Simplify expressions for easier computation.

Different Factoring Techniques

Some common methods include:

• Common factor extraction: As seen in our example, pulling out common numbers or expressions.
• Factoring quadratics: Using methods like the quadratic formula or completing the square.
• Splitting the middle term: Particularly useful when dealing with trinomials.

Factoring is pivotal because it untangles the expression, making it simpler and clearer. In our current example, once \(3x + 9\) is factored to \(3(x + 3)\), it becomes apparent that both the numerator and denominator share \(x + 3\), a common factor, simplifying the expression considerably.

Domain Restrictions

Domain restrictions are crucial to consider when working with rational expressions. In simple terms, these are conditions on the variables that prevent division by zero, a mathematical immateriality.

For our fraction \( \frac{x+3}{3x+9} \), before simplification, we must check where the denominator equals zero. After factoring, the expression becomes \( \frac{x+3}{3(x+3)} \). This tells us the shared factor in the denominator and numerator. However, setting \( x+3 = 0 \) would make the denominator zero, which is not allowed.

Why Does This Matter?

Understanding domain restrictions ensures:

• We avoid division by zero, maintaining mathematical integrity.
• We accurately describe where the expression is valid.

How to Find Domain Restrictions:

• Set the denominator equal to zero to find critical points.
• Solve the equation to discover restricted values; in our case, \(x eq -3\).

Always remember to confirm domain restrictions, especially before canceling terms. Doing so safeguards against possible errors and anomalies in your results.