Problem 17 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{n^{2}}{n^{2}+3 n+2}-\frac{1}{n^{2}+3 n+2} $$

Step-by-Step Solution

Answer

\( \frac{n - 1}{n + 2} \)

Step 1: Identify the common denominator

Notice that both fractions share the same denominator: \( n^2 + 3n + 2 \).

Step 2: Combine the fractions

Combine the fractions over the common denominator: \( \frac{n^{2}}{n^{2}+3n+2} - \frac{1}{n^{2}+3n+2} = \frac{n^{2} - 1}{n^{2}+3n+2} \).

Step 3: Simplify the numerator

Notice that the numerator \( n^2 - 1 \) can be factored using the difference of squares: \( n^2 - 1 = (n - 1)(n + 1) \).

Step 4: Factor the denominator

Factor the denominator \( n^2 + 3n + 2 \). Find the factors of 2 that add up to 3: \( (n + 1)(n + 2) \).

Step 5: Simplify the fraction

Combine your factored parts to get \( \frac{(n - 1)(n + 1)}{(n + 1)(n + 2)} \). Cancel out the common term \( (n + 1) \) to simplify to \( \frac{n - 1}{n + 2} \).

Key Concepts

Factoring Polynomials

Factoring polynomials breaks down complex expressions into simpler components. In our solution, we factor the denominator \( n^2 + 3n + 2 \). This expression is simplified by finding two numbers that multiply to give the constant term (2) and add to give the middle coefficient (3).
This leads us to the factors \( (n + 1) \) and \( (n + 2) \). Thus, \( n^2 + 3n + 2 = (n + 1)(n + 2) \).
The numerator \( n^2 - 1 \) is factored using the difference of squares, resulting in \( (n - 1)(n + 1) \).
Factoring helps in straightforward manipulation and simplification.

Combining Fractions

Combining fractions means to express them as a single fraction. Fractions can only be combined directly if they have a common denominator. In the given problem:

• We start with two fractions: \( \frac{n^2}{n^2 + 3n + 2} \) and \( \frac{1}{n^2 + 3n + 2} \).
• Since both have the same denominator \( n^2 + 3n + 2 \), we can combine them.

By placing them over a common denominator, we get:
\[ \frac{n^2 - 1}{n^2 + 3n + 2} \]Combining fractions simplifies the equation and makes it easier to further simplify and solve.

Difference of Squares

The difference of squares is a special factoring formula:\( a^2 - b^2 = (a - b)(a + b) \). In our problem:

• The numerator \( n^2 - 1 \) can be seen as the difference of squares with \( n^2 \) and \( 1 \) or \( (n)^2 - (1)^2 \).
• Using the difference of squares formula, we get \( (n - 1)(n + 1) \).

This step is crucial as it simplifies the numerator, allowing for possible cancellations with the denominator further down the simplification process.

Common Denominators

Common denominators are key when combining fractions. A common denominator is a shared multiple of the denominators of each fraction involved. In this problem,\( n^2 + 3n + 2 \) acts as the common denominator.
Using the factored form:\( (n + 1)(n + 2) \), both fractions can be rewritten simply. This allows us to easily combine and later simplify the fractions. Here's a step-by-step breakdown:

• Recognize the shared denominator in each fraction.
• Write the fractions under a single denominator.
• Simplify by reducing and canceling common factors.

This method helps in transforming complex equations into manageable ones, enabling further simplification and solution.