Problem 59 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{1}{x^{2}+x-12}-\frac{1}{x^{2}-7 x+12}+\frac{1}{x^{2}-16}$$

Step-by-Step Solution

Answer

The result is \( \frac{x - 11}{(x+4)(x-3)(x-4)} \).

Step 1: Factor the Denominators

First, we factor the denominators of each rational expression. For the first term, the denominator is \(x^2 + x - 12\), which factors to \((x + 4)(x - 3)\). For the second term, \(x^2 - 7x + 12\) factors to \((x - 3)(x - 4)\). For the last term, \(x^2 - 16\) factors to \((x - 4)(x + 4)\).

Step 2: Determine the Common Denominator

The least common denominator (LCD) for the fractions is the product of the different linear factors from the factored denominators. The common denominator is \((x + 4)(x - 3)(x - 4)\).

Step 3: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction over the common denominator:\[ \frac{1}{(x+4)(x-3)} = \frac{x-4}{(x+4)(x-3)(x-4)} \]\[ \frac{1}{(x-3)(x-4)} = \frac{x+4}{(x+4)(x-3)(x-4)} \]\[ \frac{1}{(x+4)(x-4)} = \frac{x-3}{(x+4)(x-3)(x-4)} \]

Step 4: Combine the Numerators

Now, we add or subtract the numerators from the fractions after rewriting:\[ (x - 4) - (x + 4) + (x - 3) \]

Step 5: Simplify the Expression

Combine the terms from step 4:\[ x - 4 - x - 4 + x - 3 = x - 11 \]Place the simplified expression over the common denominator:\[ \frac{x - 11}{(x+4)(x-3)(x-4)} \]

Key Concepts

Factorization

Factorization is a key technique used in algebra to make expressions simpler and easier to work with, particularly when dealing with polynomial expressions. In this exercise, the denominators of the rational expressions need to be factored to identify their linear components. Factorization helps us break down polynomials into a product of simpler terms. In our case:

• For \( x^2 + x - 12 \), we need to find two numbers that multiply to \(-12\) and add to \(1\), which are \(4\) and \(-3\). Thus, \( x^2 + x - 12 \) factors to \((x + 4)(x - 3)\).
• For \( x^2 - 7x + 12 \), we look for two numbers multiplying to \(12\) and adding to \(-7\), which are \(-3\) and \(-4\). Thus, it factors to \((x - 3)(x - 4)\).
• Lastly, \( x^2 - 16 \) is a difference of squares, factoring directly to \((x - 4)(x + 4)\).

Understanding factorization is not only critical for solving this problem but also for many areas in mathematics. It helps in solving equations, simplifying expressions, and finding common denominators.

Common Denominator

To perform addition or subtraction with rational expressions, like fractions, the denominators must be the same. This is known as finding a common denominator.
The given expressions have the following factors from their denominators:

• First expression: \((x + 4)(x - 3)\)
• Second expression: \((x - 3)(x - 4)\)
• Third expression: \((x - 4)(x + 4)\)

The least common denominator (LCD) is obtained by taking each unique factor the greatest number of times it occurs in any of the denominators. Thus, the LCD for these fractions is \((x + 4)(x - 3)(x - 4)\). Finding a common denominator allows us to securely manipulate the expressions to have a unified base, making the arithmetic operations (addition and subtraction) straightforward.

Simplifying Fractions

Simplifying fractions involves rewriting the expression in a more compact form while retaining its equivalence. When we have a common denominator for all terms, the next step is to perform operations on the numerators.
In the exercise, we rewrite each fraction with the common denominator and adjust their numerators:

• The first fraction becomes \( \frac{x-4}{(x+4)(x-3)(x-4)} \).
• The second becomes \( \frac{x+4}{(x+4)(x-3)(x-4)} \).
• The third becomes \( \frac{x-3}{(x+4)(x-3)(x-4)} \).

With a common base, we can now efficiently subtract and add the numerators:

• Combine \((x - 4) - (x + 4) + (x - 3)\) to get \(x - 4 - x - 4 + x - 3\), simplified to \(x - 11\).

Finally, place this combined numerator over the common denominator, resulting in a simplified expression: \[ \frac{x - 11}{(x+4)(x-3)(x-4)} \].Thus, simplifying fractions involves careful consideration of each component within the expression, ensuring that operations are applied correctly for accurate results.