Problem 48 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{2 x+6}{x^{2}+6 x+9}+\frac{5 x}{x^{2}-9}+\frac{7}{x-3}$$

Step-by-Step Solution

Answer

\(\frac{14x + 15}{(x+3)(x-3)}\)

Step 1: Factor the Denominators

First, we need to factor each denominator wherever possible.1. For the first term, the denominator is a quadratic: \( x^2 + 6x + 9 = (x + 3)^2 \), using the perfect square trinomial formula.2. For the second term, recognize the difference of squares: \( x^2 - 9 = (x - 3)(x + 3) \).3. The third term has a linear denominator \( x - 3 \), which is already factored.

Step 2: Simplify Each Fraction

Now, simplify each fraction if possible by canceling common factors in the numerator and denominator.1. First term: \( \frac{2x+6}{(x+3)^2} = \frac{2(x+3)}{(x+3)^2} = \frac{2}{x+3} \), after canceling \(x+3\).2. Second term: There's no factor to cancel with the numerator \( \frac{5x}{(x-3)(x+3)} \).3. Third term: Already simple, \( \frac{7}{x-3} \).

Step 3: Find the Common Denominator

Next, we look for a common denominator, which must include each distinct factor from the factored denominators:- The necessary factors are \( (x + 3)^1 \) and \( (x - 3)\).- Therefore, the least common denominator (LCD) is \((x+3)(x-3)\).

Step 4: Rewrite Each Fraction with the Common Denominator

Convert each fraction to have the common denominator:1. \( \frac{2}{x+3} = \frac{2(x-3)}{(x+3)(x-3)} \)2. \( \frac{5x}{(x-3)(x+3)} \) is already over the common denominator.3. \( \frac{7}{x-3} = \frac{7(x+3)}{(x+3)(x-3)} \)

Step 5: Combine the Fractions

Now combine the fractions over the common denominator:\[\frac{2(x-3)+5x+7(x+3)}{(x+3)(x-3)}\]Simplify the numerator:- Distribute and combine like terms:- \( 2(x-3) = 2x - 6 \)- \( 7(x+3) = 7x + 21 \)- The combined numerator is \((2x - 6) + 5x + (7x + 21) = 14x + 15\).

Step 6: Simplify the Final Expression

The final step is to write the simplified expression:\[\frac{14x + 15}{(x+3)(x-3)}\]This expression cannot be simplified further.

Key Concepts

Factoring Trinomials

Factoring trinomials is an important skill when dealing with rational expressions. A trinomial is a polynomial with three terms. Often, these trinomials can be factored into a product of two binomials.

In our problem, the first term has a quadratic trinomial in the denominator: \(x^2 + 6x + 9\). By recognizing this as a perfect square trinomial, we can factor it as \((x + 3)^2\).

• A perfect square trinomial takes the general form \(a^2 + 2ab + b^2\)
• It factors to \((a + b)^2\)

Recognizing these patterns allows us to rewrite the trinomial in a simpler, equivalent form.
Additionally, the second term has \(x^2 - 9\) as the denominator. This is a difference of squares, which factors into \((x - 3)(x + 3)\). Here’s how that pattern works:

• For two squares \(a^2 - b^2\), it factors to \((a - b)(a + b)\)
• In this case, \(x^2-9 = (x-3)(x+3)\)

Least Common Denominator

When adding or subtracting rational expressions, finding a common denominator is crucial. This common denominator must contain each distinct factor found in the denominators of all the terms.

There may be different methods to find the Least Common Denominator (LCD), but here is a straightforward way:

• Identify all unique factors in the denominators.
• Make sure the powers of each factor are high enough to cover all occurrences in the expressions.

In the exercise, the factored denominators were \((x + 3)^2\), \((x - 3)(x + 3)\), and \(x - 3\). To cover all these factors, the expression’s LCD should be \((x + 3)(x - 3)\). This covers:

• \((x + 3)^1\) as it appeared in \((x - 3)(x + 3)\)
• \((x - 3)\) as a separate term

This common denominator allows for an efficient way of rewriting each fraction so that the numerators can be combined.

Simplifying Fractions

Simplifying fractions is a vital part of working with rational expressions. This process involves reducing fractions to their simplest form by cancelling out common factors from the numerator and the denominator.

Here's a step-by-step to simplify effectively:

• Factor both the numerator and the denominator if possible.
• Cancel out any common factors.

Let's take the first fraction from the exercise: \(\frac{2x+6}{(x+3)^2}\).

• The numerator \(2x+6\) can be factored as \(2(x+3)\).
• The denominator \((x+3)^2\) has a common factor \((x+3)\).
• When you cancel \((x+3)\) from both numerator and denominator, it simplifies to \(\frac{2}{x+3}\).

The other fractions may not always have factors to cancel, as seen in the second term; however, identifying these possibilities is key. This simplification step helps in the process of combining fractions by making the expression easier to manage.