What is the difference?

[tex]\[ \frac{2x+5}{x^2-3x} - \frac{3x+5}{x^3-9x} - \frac{x+1}{x^2-9} \][/tex]

A. [tex]\(\frac{(x+5)(x+2)}{x^3-9x}\)[/tex]

B. [tex]\(\frac{(x+5)(x+4)}{x^3-9x}\)[/tex]

C. [tex]\(\frac{-2x+11}{x^3-12x-9}\)[/tex]

D. [tex]\(\frac{3(x+2)}{x^2-3x}\)[/tex]



Answer :

Let's find the difference of the fractions step-by-step to determine which of the given options it matches.

Given:

[tex]\[ f_1(x) = \frac{2x + 5}{x^2 - 3x}, \quad f_2(x) = \frac{3x + 5}{x^3 - 9x}, \quad f_3(x) = \frac{x + 1}{x^2 - 9} \][/tex]

We need to find:

[tex]\[ f(x) = f_1(x) - f_2(x) - f_3(x) \][/tex]

So, let's perform the subtraction:

1. Compute [tex]\( \frac{2x + 5}{x^2 - 3x} - \frac{3x + 5}{x^3 - 9x} \)[/tex]
2. Then subtract [tex]\(\frac{x + 1}{x^2 - 9}\)[/tex] from the result.

First, note the denominators.
[tex]\[ x^2 - 3x = x(x - 3) \][/tex]
[tex]\[ x^3 - 9x = x(x^2 - 9) = x(x - 3)(x + 3) \][/tex]

So, the Least Common Denominator (LCD) would be [tex]\( x(x - 3)(x + 3) \)[/tex] for the first two terms and we will fit the third term similarly.

Combining fractions within the same denominator:

For the first two fractions:
[tex]\[ f_1(x) = \frac{(2x + 5)(x + 3)}{x(x - 3)(x + 3)} \][/tex]
[tex]\[ f_2(x) = \frac{3x + 5}{x(x - 3)(x + 3)} \][/tex]

[tex]\[ f_1(x) - f_2(x) = \frac{(2x + 5)(x + 3) - (3x + 5)}{x(x - 3)(x + 3)} \][/tex]

Simplifying the numerator:
[tex]\[ (2x + 5)(x + 3) - (3x + 5) = 2x^2 + 6x + 5x + 15 - 3x - 5 = 2x^2 + 8x + 10 - 3x - 5 = 2x^2 + 5x + 5 \][/tex]

So:
[tex]\[ f_1(x) - f_2(x) = \frac{2x^2 + 5x + 5}{x(x - 3)(x + 3)} \][/tex]

Now, include [tex]\( \frac{x + 1}{x^2 - 9} \)[/tex] in the subtraction:
[tex]\[ f_3(x) = \frac{x + 1}{(x - 3)(x + 3)} = \frac{x + 1}{x^2 - 9} \][/tex]

Combining all fractions:
[tex]\[ f(x) = \frac{2x^2 + 5x + 5 - (x(x + 1))}{x(x - 3)(x + 3)} = \frac{2x^2 + 5x + 5 - (x^2 + x)}{x(x - 3)(x + 3)} = \frac{2x^2 + 5x + 5 - x^2 - x}{x(x - 3)(x + 3)} = \frac{x^2 + 4x + 5}{x(x - 3)(x + 3)} \][/tex]

Finally, the simplified difference is:
[tex]\[ f(x) = \frac{(x + 5)(x + 2)}{x^3 - 9x} \][/tex]

Therefore, the correct option is:
[tex]\[ \boxed{\frac{(x + 5)(x + 2)}{x^3 - 9x}} \][/tex]