Problem 38 - Algebra for College Students

Question

Perform the indicated operations, and express your answers in simplest form. $$ \frac{2 n^{2}}{n^{4}-16}-\frac{n}{n^{2}-4}+\frac{1}{n+2} $$

Step-by-Step Solution

Answer

\[ \frac{-n^3 + 3n^2 - 4n - 4}{(n^2 + 4)(n + 2)(n - 2)} \]

Step 1: Factor the Denominators

To simplify fractions, we start by factoring the denominators. For the first term, we have the expression \( n^4 - 16 \), which is a difference of squares. It can be factored as \( (n^2 + 4)(n^2 - 4) \). Notice that \( n^2 - 4 \) is also a difference of squares: \( (n + 2)(n - 2) \).Thus, \( n^4 - 16 = (n^2 + 4)(n + 2)(n - 2) \).For the second term, the denominator \( n^2 - 4 \) is already factored as \( (n + 2)(n - 2) \).The third term denominator \( n + 2 \) is already in simplest form.

Step 2: Identify the Least Common Denominator (LCD)

To subtract fractions, we need a common denominator. From the previous step, we have:- First term: \( (n^2 + 4)(n + 2)(n - 2) \)- Second term: \( (n + 2)(n - 2) \)- Third term: \( n + 2 \)The least common denominator (LCD) encompassing all parts is \( (n^2 + 4)(n + 2)(n - 2) \).

Step 3: Write Each Fraction with the LCD

Convert each term to have the LCD as a common denominator:1. First term: \( \frac{2n^2}{(n^2 + 4)(n + 2)(n - 2)} \) already has the LCD.2. Second term: \( \frac{n}{(n + 2)(n - 2)} \) needs to be multiplied by \( \frac{n^2 + 4}{n^2 + 4} \), giving: \( \frac{n(n^2 + 4)}{(n^2 + 4)(n + 2)(n - 2)} \).3. Third term: \( \frac{1}{n+2} \) needs to be multiplied by \( \frac{n^2 - 4}{n^2 - 4} \) since \( n^2 - 4 = (n + 2)(n - 2) \), giving: \( \frac{n^2 - 4}{(n^2 + 4)(n + 2)(n - 2)} \).

Step 4: Subtract and Combine the Numerators

Now we have:\[\frac{2n^2}{(n^2 + 4)(n + 2)(n - 2)} - \frac{n(n^2 + 4)}{(n^2 + 4)(n + 2)(n - 2)} + \frac{n^2 - 4}{(n^2 + 4)(n + 2)(n - 2)}\]Combine the numerators:\[ 2n^2 - n(n^2 + 4) + (n^2 - 4) \]Simplify the expression:\[ 2n^2 - n^3 - 4n + n^2 - 4 \]Combine like terms:\[ -n^3 + 3n^2 - 4n - 4 \]

Step 5: Simplify the Expression

The entire fraction becomes:\[ \frac{-n^3 + 3n^2 - 4n - 4}{(n^2 + 4)(n + 2)(n - 2)} \]Look for simplifying further, but this expression is already in its simplest form given it has no common factors between the numerator and denominator.

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler factors. This is crucial when working with rational expressions to simplify or manipulate them.
Let's consider the polynomial expressions in the original problem. The denominator of the first fraction, \( n^4 - 16 \), can be seen as a difference of squares, which is a standard form: \( a^2 - b^2 = (a - b)(a + b) \). By factoring \( n^4 - 16 \), we get \( (n^2 + 4)(n^2 - 4) \).
The expression \( n^2 - 4 \) is also a difference of squares, factoring further to \( (n + 2)(n - 2) \). This process involves identifying a quadrilateral or higher power term and breaking it down into products of binomials.
Factoring helps in recognizing common terms in different denominators, simplifying the problem of finding a common denominator later on. Each factorization step reduces the complexity of the expression and primes it for operations like addition or subtraction.

Least Common Denominator

When adding or subtracting rational expressions, it is vital to have a common denominator for all terms involved. The least common denominator (LCD) is the smallest common multiple of the denominators.
To determine the LCD for our given expressions, we review the factored forms:

• First term: \((n^2 + 4)(n + 2)(n - 2)\)
• Second term: \((n + 2)(n - 2)\)
• Third term: \(n + 2\)

The LCD must include each factor that appears in any denominator. Here, the LCD becomes \((n^2 + 4)(n + 2)(n - 2)\) as it encompasses all factors from each fraction's denominator.
This ensures that each term can be written with the same denominator, allowing for straightforward combination or subtraction of numerators.

Simplifying Expressions

Simplifying expressions is about reducing an expression to its most basic form without changing its value. Once all terms of an expression have a common denominator, you can combine them by adding or subtracting the numerators.
In our solution, after finding a common denominator, each expression was adjusted to have this LCD, allowing for the numerators to be combined:

• First, write all terms with the common denominator.
• Then, perform the operations required in the numerators.
• Finally, combine the terms: \( 2n^2 - n(n^2 + 4) + (n^2 - 4) \).

After expanding and combining like terms, the expression \(-n^3 + 3n^2 - 4n - 4\) in the numerator represents the simplified form.
It is important to express the final form in its simplest state by removing any further common factors, but in this exercise, no further simplification was possible. Keeping expressions simple avoids complications in further mathematical processes.