- Factor the denominators of each term to find the LCD \(x^3-9x\).
- Rewrite each fraction to have the common denominator \(x^3-9x\).
- Combine the numerators and simplify the resulting polynomial to \(x^2+7x+10\).
- Factor the simplified numerator to match the result: \(\boxed{\frac{(x+5)(x+2)}{x^3-9x}}\).
Explanation
- Analyzing Denominators and Identifying the LCD
We are tasked with simplifying the expression \(\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9}\).
First, we factor each of the denominators to identify the least common denominator (LCD):
- For the first fraction, \(x^2 - 3x = x(x-3)\).
- For the second fraction, \(x^3 - 9x = x(x^2-9) = x(x-3)(x+3)\).
- For the third fraction, \(x^2 - 9 = (x-3)(x+3)\).
The common denominator for all terms is \(x(x-3)(x+3) = x^3 - 9x\).
- Rewriting Fractions with the Common Denominator
Next, we rewrite each term using the LCD \(x^3-9x\):
- For the first term \(\frac{2x+5}{x(x-3)}\), we multiply the numerator and denominator by \((x+3)\):
\[\frac{(2x+5)(x+3)}{x(x-3)(x+3)} = \frac{2x^2 + 6x + 5x + 15}{x^3-9x} = \frac{2x^2 + 11x + 15}{x^3-9x}.\]
- The second term \(\frac{3x+5}{x^3-9x}\) already has the common denominator.
- For the third term \(\frac{x+1}{(x-3)(x+3)}\), we multiply the numerator and denominator by \(x\):
\[\frac{(x+1)x}{x(x-3)(x+3)} = \frac{x^2+x}{x^3-9x}.\]
- Combining and Simplifying the Numerator
Now, we combine the fractions by subtracting the numerators over the common denominator:
\[\frac{(2x^2 + 11x + 15) - (3x+5) - (x^2 + x)}{x^3-9x}\]
Simplifying the numerator:
\[(2x^2 + 11x + 15) - 3x - 5 - x^2 - x\]
Group the terms by power of \(x\):
\[(2x^2 - x^2) + (11x - 3x - x) + (15 - 5)\]
\[x^2 + 7x + 10\]
Thus, the simplified expression is \(\frac{x^2+7x+10}{x^3-9x}\).
- Factoring the Numerator and Concluding
Finally, we factor the numerator \(x^2+7x+10\) into \((x+5)(x+2)\).
Comparing this to our initial results:
\[\frac{(x+5)(x+2)}{x^3-9x}\]
This corresponds exactly to the first option. The difference is \(\boxed{\frac{(x+5)(x+2)}{x^3-9x}}\).
Examples
Simplifying rational expressions is essential in fields like electrical engineering, where you often need to combine resistances or impedances represented as complex fractions in a circuit. By finding a common denominator and simplifying the expression, engineers can reduce complicated circuit equations into a manageable form to determine the total resistance of the system. For instance, if the individual components are given by \(\frac{2x+5}{x^2-3x}\), \(\frac{3x+5}{x^3-9x}\), and \(\frac{x+1}{x^2-9}\), simplifying the total expression \(\frac{(x+5)(x+2)}{x^3-9x}\) allows for much easier analysis of the circuit's behavior.