Problem 20 - Algebra 2 and Trigonometry

Question

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{1}{x}+\frac{1}{x-2}-\frac{2}{x^{2}-2 x} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{2}{x} \), undefined for \( x = 0 \) and \( x = 2 \).

Step 1: Identify the fractions and common denominator

The given expression is \( \frac{1}{x} + \frac{1}{x-2} - \frac{2}{x^2-2x} \). Start by identifying the denominators: \( x \), \( x-2 \), and \( x(x-2) \). The expression \( x^2-2x \) factors to \( x(x-2) \). Therefore, the common denominator is \( x(x-2) \).

Step 2: Rewrite each fraction with the common denominator

Convert each fraction to have the common denominator \( x(x-2) \):1. \( \frac{1}{x} = \frac{1\cdot (x-2)}{x(x-2)} = \frac{x-2}{x(x-2)} \).2. \( \frac{1}{x-2} = \frac{1\cdot x}{x(x-2)} = \frac{x}{x(x-2)} \).3. \( \frac{2}{x^2-2x} = \frac{2}{x(x-2)} \).

Step 3: Perform the addition and subtraction

Combine the fractions:\[\frac{x-2}{x(x-2)} + \frac{x}{x(x-2)} - \frac{2}{x(x-2)}\]By combining the numerators, we get:\[\frac{(x-2) + x - 2}{x(x-2)}\]Simplify the numerator:\[(x-2) + x - 2 = 2x - 4\]Thus, the combined fraction is:\[\frac{2x-4}{x(x-2)}\]

Step 4: Simplify the expression

Factor the numerator \( 2x-4 \) as \( 2(x-2) \):\[\frac{2(x-2)}{x(x-2)}\]Cancel out the \( (x-2) \) term:\[\frac{2}{x}\]This is the simplest form of the expression.

Step 5: Identify the undefined values of the variables

The fractions are not defined if the denominator is zero, i.e., if \( x = 0 \) or \( x-2 = 0 \). Solve for these values:1. \( x = 0 \)2. \( x-2 = 0 \) implies \( x = 2 \)Therefore, the expression is undefined for \( x = 0 \) and \( x = 2 \).

Key Concepts

Common Denominator

When it comes to fraction addition and subtraction, finding a common denominator is key. This is because you can only directly add or subtract fractions when they have the same denominator. A common denominator is the smallest multiple that all denominators share.
In the expression \( \frac{1}{x} + \frac{1}{x-2} - \frac{2}{x^2-2x} \), the denominators are \( x \), \( x-2 \), and \( x(x-2) \). Notice how the denominator \( x(x-2) \) is, in fact, the least common multiple of \( x \) and \( x-2 \).

• To calculate a common denominator, consider the factors of each denominator.
• Determine any shared factors among the denominators.
• Multiply these factors to find the smallest shared denominator, in this case, \( x(x-2) \).

This allows you to rewrite each fraction with identical denominators, enabling you to add or subtract them directly after manipulating the numerators.

Simplification of Rational Expressions

Simplifying rational expressions often involves manipulating the numerator and denominator to write the expression in its simplest form. After finding a common denominator, it's all about simplifying the resulting expression.

• Combine the numerators of the fractions now that the denominators are the same. For our example: \( \frac{x-2}{x(x-2)} + \frac{x}{x(x-2)} - \frac{2}{x(x-2)} \) can be combined to form: \( \frac{2x - 4}{x(x-2)} \).
• Factor any common terms in the numerator to its simplest factorable form. In this instance, \( 2x - 4 \) can be rewritten as \( 2(x - 2) \).
• Cancel out any identical terms found in both the numerator and the denominator. For the example: \( \frac{2(x-2)}{x(x-2)} \) becomes \( \frac{2}{x} \) as \( (x-2) \) cancels out.

Through these steps, you ensure that the rational expression is in its most reduced form, which simplifies analysis and further calculations.

Undefined Values in Rational Expressions

Understanding when a rational expression is undefined is crucial in algebra to avoid errors in calculations. A rational expression is undefined wherever its denominator is zero. An undefined value often causes problems in calculations because division by zero is not defined mathematically.
For the expression given, \( \frac{2}{x} \), we need to determine values for \( x \) that would result in a zero denominator.

• Identify each part of the denominators from the original fractions before simplification: \( x(x-2) \).
• Find the values of \( x \) that cause the denominator terms to equate to zero. Thus, solve \( x = 0 \) and \( x-2 = 0 \).
• This gives us \( x = 0 \) and \( x = 2 \) as points where the original expression is undefined.

These insights ensure you know the boundaries of applying the expression properly, avoiding undefined or erroneous computations.