Problem 22 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{1}{h^{2}-4 h-5}+\frac{h}{h^{2}-4 h-5}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{h - 5}\).

Step 1: Identify the Denominators

Observe that both fractions in the expression have the same denominator: \(h^2 - 4h - 5\). Since the denominators are the same, we can add the numerators directly.

Step 2: Combine the Numerators

Add the numerators together: \(1 + h\). This gives us one single fraction over the shared denominator: \(\frac{1 + h}{h^2 - 4h - 5}\).

Step 3: Factor the Denominator

To simplify the fraction further, factor the quadratic expression in the denominator: \(h^2 - 4h - 5\). We need to find two numbers that multiply to -5 and add to -4. They are -5 and 1. So, the factorization is: \((h - 5)(h + 1)\).

Step 4: Cancel Common Factors

Check if there are any common factors between the numerator and denominator. The numerator is \(h + 1\), which is a factor in the denominator. Thus, we can cancel \(h + 1\) from both the numerator and the denominator: \(\frac{1 + h}{(h + 1)(h - 5)} = \frac{1}{h - 5}\), after cancellation.

Step 5: Conclusion

The original expression \(\frac{1}{h^{2}-4 h-5} + \frac{h}{h^{2}-4 h-5}\) simplifies to \(\frac{1}{h - 5}\) after adding the numerators and canceling the common factor in the numerator and the denominator.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Think of them like fractions you already know, but instead of whole numbers, you have expressions with variables. Just like any other fraction, rational expressions can be added, subtracted, multiplied, or divided. However, these operations often require steps like finding common denominators or factoring polynomials.

In an algebraic context, an expression becomes rational when it can be represented as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero. In our problem,

• Both numerators \(1\) and \(h\) are simple polynomials.
• The shared denominator \(h^2 - 4h - 5\) is also a polynomial.

Understanding the structure of rational expressions helps break down complex algebraic tasks into manageable parts.

It forms a solid foundation for simplifying, factoring, and performing operations, just like with simpler numeric fractions.

Factoring Quadratics

Factoring quadratics is a crucial skill when handling algebraic fractions. The process involves rewriting a quadratic expression as a product of its factors. The goal is to make complex expressions simpler or to identify and cancel out common terms later.

To factor a quadratic like \(h^2 - 4h - 5\), follow these steps:

• Identify two numbers that multiply to the constant term (-5) and add to the linear coefficient (-4). Here they are -5 and 1.
• Rewrite the expression as \( (h-5)(h+1) \).

Factoring is often the key to simplifying rational expressions by allowing cancellation of common factors between the numerator and denominator.

This process not only simplifies expressions but also helps solve equations involving quadratics by setting each factor to zero. In our problem, recognizing that \(h + 1\) in the numerator is also a factor in the denominator was crucial for simplification.

Simplifying Fractions

Simplifying fractions, whether numeric or algebraic, aims to reduce them to their simplest form. This involves canceling common factors in both the numerator and the denominator. For rational expressions, after factoring the components, look for identical terms that appear in both parts.

In the exercise, once the quadratic in the denominator is factored, it becomes clear that \(h + 1\) appears in both the numerator and part of the denominator, \( (h+1)(h-5) \). By canceling \(h + 1\), we significantly simplify the expression to \( \frac{1}{h - 5} \).

Tips for simplifying:

• Always factor both the numerator and the denominator if possible.
• Cancel out only when terms are exactly the same in both parts.
• Keep in mind any restrictions, such as values that would make the denominator zero, since these would invalidate the expression.

Understanding how to simplify rational expressions is critical, as it allows you to work with more complex equations and functions easily, focusing only on the essential elements.