Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam Simplify: \(\frac{3}{x^{2}-9}-\frac{2}{x+3}\)

Short answer

\( \frac{9-2x}{(x-3)(x+3)} \)

Step-by-step solution

Identify the Problem

The expression to simplify is: \[ \frac{3}{x^{2}-9}-\frac{2}{x+3} \]

Factor the Denominator

Identify that the first term's denominator \(x^{2}-9\) is a difference of squares. Factor it: \[ x^{2} - 9 = (x-3)(x+3) \] Now, the expression becomes: \[ \frac{3}{(x-3)(x+3)} -\frac{2}{x+3} \]

Combine the Fractions

Find a common denominator, which is \((x-3)(x+3)\). Adjust the second fraction to have this common denominator: \[ \frac{2}{x+3} = \frac{2(x-3)}{(x-3)(x+3)} \] Thus, the expression becomes: \[ \frac{3}{(x-3)(x+3)} - \frac{2(x-3)}{(x-3)(x+3)} \]

Simplify the Numerator

Combine the fractions by subtracting the numerators over the common denominator: \[ \frac{3 - 2(x-3)}{(x-3)(x+3)} \] Simplify the numerator: \[ 3 - 2(x-3) = 3 - 2x + 6 = 9 - 2x \] Thus, the expression becomes: \[ \frac{9-2x}{(x-3)(x+3)} \]

Final Simplified Form

Write the final simplified form of the expression: \[ \frac{9-2x}{(x-3)(x+3)} \]

Key concepts

Difference of Squares

The difference of squares is a fundamental algebraic concept you will come across often. It states that any expression of the form \( a^2 - b^2 \) can be broken into the product of two binomials: \((a+b)(a-b)\).
In the original exercise, the term \( x^2 - 9 \) is recognized as a difference of squares. Here, \(a\) is \( x \) and \(b\) is 3. So, \( x^2 - 9 \) can be factored into \( (x-3)(x+3)\). Recognizing the difference of squares makes simplification of algebraic fractions much easier.
Whenever you see \( a^2 - b^2 \), think immediately of \( (a+b)(a-b)\). This simple transformation is a powerful tool for simplifying complex expressions in algebra.

Common Denominators

When adding or subtracting fractions, having a common denominator is crucial. Without it, you can't combine the fractions effectively.
In this exercise, after factoring the denominator of the first term, we see that \( \frac{3}{(x-3)(x+3)} \) and \(\frac{2}{x+3} \) have different denominators. To combine these, we need a common denominator.
The common denominator here is \(( x-3 )( x+3 )\), which is essentially the original denominator from the first fraction but applied to both terms. This allows us to rewrite the second fraction with the common denominator: \( \frac{2}{x+3} = \frac{2(x-3)}{(x-3)(x+3)}\).
Remember, when finding a common denominator, always look for the least common multiple, which includes all unique factors from each denominator.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form. This often involves factoring numerators and denominators and then cancelling out common factors if possible.
In our example, after finding the common denominator, we combine the fractions: \(\frac{3}{(x-3)(x+3)} - \frac{2(x-3)}{(x-3)(x+3)}\).
The next step is to combine the numerators and write them over the common denominator: \(\frac{3 - 2(x-3)}{(x-3)(x+3)}\). Once we do the arithmetic in the numerator: \(3 - 2(x-3)\), it simplifies to \(9 - 2x\).
Therefore, the final step gives us the simplified fraction: \(\frac{9-2x}{(x-3)(x+3)}\). This demonstrates how each step of simplification builds on the previous ones.

Numerator and Denominator

In any fraction, the top part is called the numerator, and the bottom part is called the denominator. To simplify a fraction, you often need to work both with the numerator and the denominator.
In the given problem, we have two fractions: \(\frac{3}{(x-3)(x+3)} \) and \(\frac{2}{x+3} \).
The numerator and denominator of each fraction play unique roles in simplification. We first focus on the denominators. By factoring and finding a common denominator, we set the stage for combining the two fractions.
Next, we turn our attention to the numerators. By performing operations such as distributing and combining like terms, we simplify the new numerator: \( 3 - 2(x-3) = 9 - 2x \).
In essence, understanding how to manipulate both the numerator and the denominator is crucial for simplifying fractions effectively.