Problem 17 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{1}{x^{2}-6 x}-\frac{1}{x^{2}+6 x} $$

Step-by-Step Solution

Answer

\( \frac{12}{x(x-6)(x+6)} \) is the simplified result.

Step 1: Find a Common Denominator

To subtract the fractions \( \frac{1}{x^2-6x} \) and \( \frac{1}{x^2+6x} \), we need to find a common denominator. The given denominators are \( x^2-6x \) and \( x^2+6x \). We can factor these expressions: \( x^2-6x = x(x-6) \) and \( x^2+6x = x(x+6) \). Thus, the common denominator is \( x(x-6)(x+6) \).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite the first fraction: \( \frac{1}{x^2-6x} = \frac{1}{x(x-6)} \). To have the common denominator \( x(x-6)(x+6) \), multiply the numerator and denominator by \( (x+6) \): \( \frac{1 \cdot (x+6)}{x(x-6)(x+6)} = \frac{x+6}{x(x-6)(x+6)} \).Similarly, rewrite the second fraction: \( \frac{1}{x^2+6x} = \frac{1}{x(x+6)} \). Multiply the numerator and denominator by \( (x-6) \): \( \frac{1 \cdot (x-6)}{x(x+6)(x-6)} = \frac{x-6}{x(x-6)(x+6)} \).

Step 3: Subtract the Fractions

Now, subtract the expressions over the common denominator:\[\frac{x+6}{x(x-6)(x+6)} - \frac{x-6}{x(x-6)(x+6)} = \frac{(x+6) - (x-6)}{x(x-6)(x+6)}\]This simplifies to:\[\frac{x+6-x+6}{x(x-6)(x+6)} = \frac{12}{x(x-6)(x+6)}\]

Step 4: Simplify the Result

The expression \( \frac{12}{x(x-6)(x+6)} \) is already in its simplest form. There are no common factors between the numerator and the denominator other than 1.

Key Concepts

Common Denominator

When performing operations with algebraic fractions, like addition or subtraction, we first need to find a common denominator. This process is similar to working with numerical fractions. The common denominator is essentially a shared base that both fractions can be rewritten to use, making it possible to perform arithmetic operations.

In our given exercise, we have two fractions: \( \frac{1}{x^2-6x} \) and \( \frac{1}{x^2+6x} \). The denominators here are polynomial expressions. To find a common denominator, you'll first factor these expressions. For \( x^2-6x \), we factor it to get \( x(x-6) \), and for \( x^2+6x \), it becomes \( x(x+6) \). The common denominator, therefore, is the product of these distinct factors: \( x(x-6)(x+6) \). This common denominator captures all unique elements from both denominators.

Finding a common denominator can be thought of as forming a new, inclusive baseline that caters to both fractions involved in your operation. This step is crucial for allowing the subtraction or addition of fractions, as it creates compatible terms for direct arithmetic.

Simplification of Expressions

Simplifying expressions involves reducing them to their most concise form. This step is crucial in mathematics because it aids in understanding, solving, and interpreting results efficiently. After achieving a common denominator, simplifying each fraction separately allows easier manipulation and cleaner results.

For example, in our exercise, after finding the common denominator \( x(x-6)(x+6) \), each fraction is rewritten. This is achieved by adjusting the numerators. The first fraction becomes \( \frac{x+6}{x(x-6)(x+6)} \) and the second \( \frac{x-6}{x(x-6)(x+6)} \). These rewritten forms maintain equivalency to the original fractions but allow subtraction owing to the shared denominator.

This step involves multiplying the numerator and denominator by missing factors, essentially ensuring the fractions are expressed in terms of the common denominator uniformly. Simplification is powerful for making complex expressions more manageable and sometimes reveals new insights about the solution.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. They are similar to numerical fractions but involve algebraic expressions. When dealing with rational expressions, key operations include finding common denominators, simplifying expressions, and ensuring the result is in the simplest form.

Rational expressions require careful handling, especially when simplifying and performing arithmetic operations. In our exercise involving \( \frac{1}{x^2-6x} - \frac{1}{x^2+6x} \), the objective was to perform subtraction. Each component of the expression behaved like a standard algebraic entity, adhering to polynomial factorization rules and simplification principles.

The outcome of these rational operations is \( \frac{12}{x(x-6)(x+6)} \), where no further simplification is possible. Always check if there are factors in common across the numerator and denominator that could allow further simplification.

• Use common denominators for arithmetic operations.
• Factor polynomials to simplify.
• Simplify results to clearest forms.

Understanding rational expressions expands algebraic manipulation skills, vital for advanced math and real-world problem-solving.