Problem 26 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{x}{x^{2}-1}+\frac{3}{x^{2}+5 x+4} $$

Step-by-Step Solution

Answer

\(\frac{x^2 + 7x - 3}{(x-1)(x+1)(x+4)}\)

Step 1: Factor Each Denominator

First, factor the denominators of each fraction. The first denominator \(x^2 - 1\) can be factored as \((x-1)(x+1)\) because it is a difference of squares. The second denominator \(x^2 + 5x + 4\) can be factored as \((x+1)(x+4)\) by finding two numbers that multiply to 4 and add to 5.

Step 2: Find a Common Denominator

Identify the least common denominator (LCD) for the two fractions. The LCD must include each unique factor from both denominators: \((x-1)(x+1)(x+4)\).

Step 3: Rewrite Each Fraction with the LCD

Rewrite each fraction with the common denominator. Multiply the numerator and denominator of the first fraction by \((x+4)\), and multiply the numerator and denominator of the second fraction by \((x-1)\). This yields: \(\frac{x(x+4)}{(x-1)(x+1)(x+4)}\) and \(\frac{3(x-1)}{(x-1)(x+1)(x+4)}\).

Step 4: Combine the Fractions

Now add the fractions since they have the same denominator. Combine their numerators: \(\frac{x(x+4) + 3(x-1)}{(x-1)(x+1)(x+4)}\).

Step 5: Simplify the Numerator

Expand and simplify the numerator: \(x(x+4) + 3(x-1) = x^2 + 4x + 3x - 3 = x^2 + 7x - 3\).

Step 6: Write the Final Expression

The final expression, in simplest form, is: \(\frac{x^2 + 7x - 3}{(x-1)(x+1)(x+4)}\).

Key Concepts

Factoring Polynomials

When working with algebraic fractions, factoring the polynomials in the denominator is often the key to simplifying the problem. In our exercise, we begin by factoring two expressions: \(x^2 - 1\) and \(x^2 + 5x + 4\). These denominators are not random; they contain hidden structures that can be revealed through factoring.

• For the first term \(x^2 - 1\), recognize it as a difference of squares. This pattern, \(a^2 - b^2 = (a-b)(a+b)\), allows us to rewrite it as \((x-1)(x+1)\).
• The second term, \(x^2 + 5x + 4\), is a trinomial. The goal is to factor it into a pair of binomials. Here, you look for numbers that multiply to 4 and add to 5, which are \(1\) and \(4\), giving us \((x+1)(x+4)\).

Once factors are identified, simplifying the problem becomes easier. Recognizing these patterns will empower you to tackle similar problems in the future.

Common Denominator

Finding a common denominator is crucial when adding or subtracting algebraic fractions, similar to how you find a common denominator in numerical fractions. A common denominator allows us to combine fractions into one expression fundamentally easier to handle.
To find the least common denominator (LCD) for the fractions \(\frac{x}{x^2-1}+\frac{3}{x^2+5x+4}\), observe all the unique factors from both denominators:

• The first fraction's denominator \((x-1)(x+1)\)
• The second fraction\'s denominator \((x+1)(x+4)\)

The LCD must include each factor at least once without repetition, resulting in \((x-1)(x+1)(x+4)\). Now both fractions can be rewritten to have this common denominator, enabling addition and simplification.

Simplifying Expressions

Once both fractions share a common denominator, the next step is to combine and simplify the resulting expression. This practice essentially includes finding a simple equivalent of the algebraic expression, free from unnecessary clutter.
For our expression, rewrite each fraction to have the common denominator \((x-1)(x+1)(x+4)\):

• Multiply the first fraction's numerator and denominator by \((x+4)\), yielding \(\frac{x(x+4)}{(x-1)(x+1)(x+4)}\).
• Multiply the second fraction by \((x-1)\), giving \(\frac{3(x-1)}{(x-1)(x+1)(x+4)}\).

Now, with like denominators, add the fractions, leading to \(\frac{x(x+4) + 3(x-1)}{(x-1)(x+1)(x+4)}\). Simplifying the numerator involves expanding and combining like terms: \(x^2+4x+3x-3\) simplifies to \(x^2+7x-3\). The final expression in its simplest form is then \(\frac{x^2+7x-3}{(x-1)(x+1)(x+4)}\). This process of simplification helps in interpreting the expression more clearly.