Problem 34 - Algebra 2

Question

Simplify each expression. Under what conditions is \(\frac{x-4}{(x+5)(x-1)}\) undefined?

Step-by-Step Solution

Answer

The expression is undefined when \(x = -5\) or \(x = 1\).

Step 1: Understanding the Expression

The given expression is \( \frac{x-4}{(x+5)(x-1)} \). We are asked to determine when this expression becomes undefined. A rational expression becomes undefined when the denominator equals zero.

Step 2: Finding Undefined Conditions

To find when the denominator equals zero, set \((x+5)(x-1) = 0\). Solve this equation for \(x\):

Step 3: Solving for Zero

Set each factor of the denominator to zero: 1. \(x+5=0\) 2. \(x-1=0\)Solve these equations to find \(x\) values that make each factor zero.

Step 4: First Denominator Solution

Solving \(x+5=0\):Subtract 5 from both sides to get \(x = -5\).

Step 5: Second Denominator Solution

Solving \(x-1=0\):Add 1 to both sides to get \(x = 1\).

Step 6: Undefined Conditions

The expression \( \frac{x-4}{(x+5)(x-1)} \) is undefined when \(x = -5\) or \(x = 1\), as these are the values that make the denominator zero.

Key Concepts

Undefined Rational Expressions

In mathematics, a rational expression is a fraction where both its numerator and denominator are polynomials. However, rational expressions become undefined when the denominator is zero. This is because division by zero is not allowed in mathematics, leading to undefined values. For instance, with the rational expression \(\frac{x-4}{(x+5)(x-1)}\), it is essential to identify the values of \(x\) that make the expression undefined. By solving the denominator's equation, we determine when it turns to zero. If this happens, the entire rational expression becomes undefined, and calculations cannot proceed unless we exclude these specific values of \(x\).

Recognizing these undefined scenarios helps in simplifying expressions and solving equations confidently. Ensuring the denominator stays non-zero ensures clear and accurate results.

Denominator

The denominator in a rational expression plays a crucial role. It's the polynomial at the bottom part of the fraction, and determining its value is key to ensure the expression remains defined.
In the expression \(\frac{x-4}{(x+5)(x-1)}\), the denominator is \((x+5)(x-1)\). To check when it becomes problematic, set it equal to zero: \((x+5)(x-1) = 0\). Solving this tells us the potential values of \(x\) that can make the expression undefined.

Understanding and calculating the denominator helps immensely in algebra, particularly in:

• Finding undefined values
• Simplifying expressions
• Determining domain restrictions in functions involving variables

Mastering this concept clears a major hurdle in algebra and aids in accurate mathematical manipulation.

Solving Equations

Solving equations is a foundational skill in algebra that involves finding the values of unknown variables. In our example with \((x+5)(x-1) = 0\), we seek to determine which \(x\) values will make the equation true. This process, called factoring, breaks down complicated expressions into simpler parts.
By individually setting each factor equal to zero, we perform a basic step:

• Set \(x+5=0\)
• Set \(x-1=0\)

Each of these smaller equations is solved separately for \(x\) to find values making the original denominator zero. This approach simplifies complex equations, isolating variables and making problem-solving more manageable. With these skills, handling algebraic equations becomes significantly more straightforward.

Zero Product Property

The Zero Product Property is a critical concept in algebra. It states that if a product of two or more factors equals zero, then at least one of the factors must be zero. This principle makes solving equations like \((x+5)(x-1) = 0\) easy. By applying the property, we can split the equation into:

• \(x+5 = 0\), leading to \(x = -5\)
• \(x-1 = 0\), giving \(x = 1\)

These solutions indicate where the rational expression becomes undefined. The Zero Product Property simplifies calculations, allowing a clear path to solutions. It is essential for solving quadratic equations and polynomial expressions efficiently. Embracing this concept enhances problem-solving skills and boosts confidence in handling variables and factors in algebraic equations.