Problem 85 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{5}{x^{2}-25}+\frac{4}{x^{2}-11 x+30}-\frac{3}{x^{2}-x-30}$$

Step-by-Step Solution

Answer

\(\frac{6x + 5}{(x-5)(x+5)(x-6)}\)

Step 1: Factor the Denominators

Start by factoring the denominator of each fraction. The denominators \(x^{2}-25\), \(x^{2}-11x+30\), and \(x^{2}-x-30\) can be factored following the rules of factoring quadratic polynomials. Factored forms are \( (x-5)(x+5) \), \( (x-6)(x-5) \) and \( (x-6)(x+5) \) respectively.

Step 2: Determine the Least Common Denominator (LCD)

The least common denominator should be the one that can be divided evenly by each of the denominators. Observing the factored forms, it's clear that the LCD is \( (x-5)(x+5)(x-6) \).

Step 3: Convert Each Fraction to Have the LCD as the Denominator

To convert each fraction, multiply the top and bottom by what's missing from the LCD. So, \(\frac{5}{(x-5)(x+5)}\) becomes \(\frac{5(x-6)}{(x-5)(x+5)(x-6)}\), \(\frac{4}{(x-5)(x-6)}\) becomes \(\frac{4(x+5)}{(x-5)(x+5)(x-6)}\), and \(\frac{3}{(x+5)(x-6)}\) becomes \(\frac{3(x-5)}{(x-5)(x+5)(x-6)}\).

Step 4: Perform the Addition and Subtraction

Now, we can perform the addition and subtraction: \(\frac{5(x-6)}{(x-5)(x+5)(x-6)} + \frac{4(x+5)}{(x-5)(x+5)(x-6)} - \frac{3(x-5)}{(x-5)(x+5)(x-6)}\). Since each expression has the same LCD, simply combine like terms from each of the numerators.

Step 5: Simplify the Fraction

Finally, combine like terms and simplify your fraction: \(\frac{5x - 30 + 4x + 20 - 3x + 15}{(x-5)(x+5)(x-6)}\). This simplifies to \(\frac{6x + 5}{(x-5)(x+5)(x-6)}\).