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Chapter 9: Problem 39

In Exercises \(1-62,\) perform the indicated operations and simplify your answer as completely as possible. $$\frac{3}{x^{2}-16}-\frac{3}{2 x^{2}+8 x}$$

Short Answer

Expert verified
\(\frac{3}{2x(x - 4)}\)

Step by step solution

01

- Factor the Denominators

First, factor the denominators of each fractional term. For the first term, the denominator is: \(x^{2} - 16 = (x + 4)(x - 4)\) For the second term, the denominator is: \(2x^{2} + 8x = 2x(x + 4)\)
02

- Find the Common Denominator

The common denominator is the least common multiple of \((x + 4)(x - 4)\) and \(2x(x + 4)\). This is: \((x + 4)(x - 4) \times 2x = 2x(x + 4)(x - 4)\)
03

- Rewrite Each Fraction

Rewrite each fraction with the common denominator: \(\frac{3}{(x + 4)(x - 4)} = \frac{3 \times 2x}{2x(x + 4)(x - 4)} = \frac{6x}{2x(x + 4)(x - 4)}\) \(\frac{3}{2x(x + 4)} = \frac{3(x - 4)}{2x(x + 4)(x - 4)} = \frac{3x - 12}{2x(x + 4)(x - 4)}\)
04

- Combine the Fractions

Combine the fractions by subtracting the numerators: \(\frac{6x - (3x - 12)}{2x(x + 4)(x - 4)} = \frac{6x - 3x + 12}{2x(x + 4)(x - 4)}\)
05

- Simplify the Expression

Simplify the numerator: \(\frac{3x + 12}{2x(x + 4)(x - 4)}\). Factor out the 3: \(\frac{3(x + 4)}{2x(x + 4)(x - 4)}\)
06

- Cancel Common Factors

Cancel the common factor of \(x + 4\) in the numerator and denominator: \(\frac{3}{2x(x - 4)}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring denominators
Factoring denominators is a crucial skill in working with algebraic fractions. This step simplifies the process of finding a common denominator and combining fractions. In our example, let's look at the denominators:
  • The first fraction has a denominator of \(x^2 - 16\). This is a difference of squares, which can be factored as \((x + 4)(x - 4)\).
  • The second fraction's denominator is \(2x^2 + 8x\). Here, we can factor out the greatest common factor, giving us \(2x(x + 4)\).
By transforming these expressions into their factored forms, we can simplify the steps needed to find common denominators and combine fractions.
Common denominators
Having a common denominator is necessary for adding or subtracting fractions. In our example, we need to find the least common multiple (LCM) of the factored denominators:
\[(x + 4)(x - 4)\] and \[2x(x + 4)\]
The LCM is the product of the highest powers of each unique factor appearing in the denominators. This gives us:
\[2x(x + 4)(x - 4)\]
By rewriting both fractions with this common denominator, we can then perform subtraction or addition more easily. Here鈥檚 how you do it:
  • Rewriting the first fraction:
    \( \frac{3}{(x + 4)(x - 4)} = \frac{3 \times 2x}{2x(x + 4)(x - 4)} = \frac{6x}{2x(x + 4)(x - 4)} \)
  • Rewriting the second fraction:
    \( \frac{3}{2x(x + 4)} = \frac{3(x - 4)}{2x(x + 4)(x - 4)} = \frac{3x - 12}{2x(x + 4)(x - 4)} \)
Simplifying rational expressions
Simplifying rational expressions involves combining like terms and reducing expressions to their simplest form. For our example:
  • Subtract the numerators of the rewritten fractions:
    \( \frac{6x - (3x - 12)}{2x(x + 4)(x - 4)} = \frac{6x - 3x + 12}{2x(x + 4)(x - 4)} \)
  • Simplify the numerator:
    \( \frac{3x + 12}{2x(x + 4)(x - 4)} \)
  • Factor out any common factors in the numerator:
    \( \frac{3(x + 4)}{2x(x + 4)(x - 4)} \)
  • Cancel out any common factors in the numerator and denominator:
    \( \frac{3}{2x(x - 4)} \)
By doing this, we achieve the simplest form of the rational expression, making it easier to understand and work with in future problems.

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