- Identify the common denominator as \(x^3-9x\) by factoring all denominators.
- Rewrite each fraction to have this common denominator.
- Combine and subtract the terms to obtain the numerator \(x^2+7x+10\).
- Factor the resulting numerator to get the final form \(\frac{(x+5)(x+2)}{x^3-9x}\), which is \(\boxed{\frac{(x+5)(x+2)}{x^3-9x}}\).
Explanation
- Analyzing Denominators and Finding the LCD
To simplify the given expression, we first analyze the denominators of each fraction:
- The first denominator is \(x^2-3x = x(x-3)\).
- The second denominator is \(x^3-9x = x(x^2-9) = x(x-3)(x+3)\).
- The third denominator is \(x^2-9 = (x-3)(x+3)\).
The least common denominator (LCD) for these three fractions is \(x(x-3)(x+3) = x^3-9x\). We will now rewrite each fraction with this common denominator to perform the subtraction.
- Rewriting Fractions with a Common Denominator
We rewrite each term using the LCD \(x(x-3)(x+3)\):
- For \(\frac{2x+5}{x(x-3)}\):
\[\frac{2x+5}{x(x-3)} = \frac{(2x+5)(x+3)}{x(x-3)(x+3)} = \frac{2x^2+6x+5x+15}{x^3-9x} = \frac{2x^2+11x+15}{x^3-9x}\]
- For the second term \(\frac{3x+5}{x^3-9x}\), no change is needed:
\[\frac{3x+5}{x^3-9x} = \frac{3x+5}{x^3-9x}\]
- For \(\frac{x+1}{(x-3)(x+3)}\):
\[\frac{x+1}{(x-3)(x+3)} = \frac{(x+1)x}{x(x-3)(x+3)} = \frac{x^2+x}{x^3-9x}\]
- Performing the Subtraction
Now, subtract the terms:
\[\frac{2x^2+11x+15}{x^3-9x} - \frac{3x+5}{x^3-9x} - \frac{x^2+x}{x^3-9x}\]
Combining them into a single numerator:
\[\frac{(2x^2+11x+15) - (3x+5) - (x^2+x)}{x^3-9x}\]
Expanding the terms in the numerator:
\[2x^2+11x+15 - 3x - 5 - x^2 - x\]
Grouping like terms:
\[(2x^2 - x^2) + (11x - 3x - x) + (15 - 5)\]
\[x^2 + 7x + 10\]
- Final Simplification and Conclusion
The simplified expression is \(\frac{x^2+7x+10}{x^3-9x}\).
We can factor the numerator \(x^2+7x+10\) as \((x+5)(x+2)\).
Thus, the final simplified expression is:
\[\frac{(x+5)(x+2)}{x^3-9x}\]
This result matches the first option.
Examples
Rational expressions like the one simplified here are foundational in engineering and physics, where complex systems are modeled using ratios of variables. For example, in electrical circuit analysis, calculating equivalent resistances or transfer functions often involves manipulating rational expressions of the frequency variable. Mastering these simplifications allows engineers to reduce complex models into manageable forms, ensuring efficient system design and analysis.