Problem 48 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{x+3}{x-3}+\frac{x-3}{x+3} $$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{x+3}{x-3} + \frac{x-3}{x+3}\) is \(\frac{2x^2+18}{x^2-9}\)

Step 1: Identify the Lowest Common Denominator

The current denominators of the fractions are \(x-3\) and \(x+3\), their product \( (x-3)(x+3) = x^2 - 9\) would be the least common denominator.

Step 2: Rewrite the fractions with the common denominator

Rewrite each fraction so that they both have the common denominator \(x^2 - 9\). This gives \(\frac{(x+3)(x+3)}{x^2 - 9} + \frac{(x-3)(x-3)}{x^2 - 9}\)

Step 3: Simplify the numerators

Simplify the numerators to get \(\frac{x^2+6x+9}{x^2-9} + \frac{x^2-6x+9}{x^2-9}\)

Step 4: Combine the fractions

As the fractions have the same denominator, they can be combined as follows \(\frac{x^2+6x+9 + x^2-6x+9}{x^2-9}\)

Step 5: Simplify the combined fraction

Further simplification gives \(\frac{2x^2+18}{x^2-9}\)

Key Concepts

Least Common Denominator

In math, when you're adding or subtracting fractions, you need a common baseline to work from, and that's where the Least Common Denominator (LCD) comes in. This is the smallest expression that all denominators can divide into evenly, and it ensures that fractions share the same denominator so they can be easily added or subtracted.
For rational expressions involving variables, finding the LCD might involve factoring.

• First, always identify the denominators of the given fractions. For example, in the problem we're discussing, \(x-3\) and \(x+3\) are the denominators.
• To get the LCD, multiply these denominators together: \( (x-3)(x+3) = x^2 - 9\).

This expression is the least common denominator, as it's the smallest polynomial that encompasses both given denominators. Using the LCD makes it simpler to manage and unify the fractions in its simplest form.

Fraction Addition

Once you've established the common denominator, it's time to add the fractions. Adding fractions isn't much different from adding numbers, but requires careful adjustment of the numerators because they must share the denominator.
Begin by rewriting each fraction with the newly found least common denominator:

• The first fraction becomes \(\frac{(x+3)(x+3)}{x^2 - 9}\) by multiplying the numerator and denominator by each other's denominator.
• Similarly, the second fraction is rewritten as \(\frac{(x-3)(x-3)}{x^2 - 9}\).

With both fractions now expressed with the same denominator, the next step is to add their numerators while retaining the common denominator:

• This results in \(\frac{(x^2+6x+9) + (x^2-6x+9)}{x^2-9}\).

The process is straightforward because the denominator remains the same, so you only need to focus on resolving and simplifying the expressions in the numerator.

Polynomial Simplification

Simplifying polynomials is what makes the final expression as clean and understandable as possible. To do this, focus on the expressions in the numerators.
In our example, there's a lot of action happening in the numerator that can be tidied up:

• After expanding and then adding terms, our numerator is \(x^2 + 6x + 9 + x^2 - 6x + 9\).
• You can immediately combine like terms: \(x^2 + x^2 = 2x^2\), and the linear terms with opposite coefficients, \(6x\) and \(-6x\), cancel each other out.
• The constants add up as \(9 + 9 = 18\).

Thus, the simplified expression over the common denominator is \(\frac{2x^2 + 18}{x^2 - 9}\).
The goal of polynomial simplification is to ensure the resulting expression is as straightforward as possible, which facilitates easier interpretation and more efficient calculations.