What is the difference?

[tex]\[
\frac{x}{x^2 - 2x - 15} - \frac{4}{x^2 + 2x - 35}
\][/tex]

A. [tex]\(\frac{x^2 + 3x + 12}{(x-3)(x-5)(x+7)}\)[/tex]

B. [tex]\(\frac{x(x+3-12)}{(x+3)(x-5)(x+7)}\)[/tex]

C. [tex]\(\frac{x^2 + 3x + 12}{(x+3)(x-5)(x+7)}\)[/tex]

D. [tex]\(\frac{x^2 + 3x - 12}{(x+3)(x-5)(x+7)}\)[/tex]



Answer :

To find the difference between the given functions [tex]\(\frac{x}{x^2-2x-15}\)[/tex] and [tex]\(\frac{4}{x^2+2x-35}\)[/tex], we need to determine a common denominator first and then combine the fractions. Let’s simplify and analyze the given expressions step by step.

1. Factor the Denominators:

The first expression is [tex]\(\frac{x}{x^2 - 2x - 15}\)[/tex]. The denominator can be factored as:
[tex]\[ x^2 - 2x - 15 = (x - 5)(x + 3) \][/tex]

The second expression is [tex]\(\frac{4}{x^2 + 2x - 35}\)[/tex]. The denominator can be factored as:
[tex]\[ x^2 + 2x - 35 = (x + 7)(x - 5) \][/tex]

2. Rewrite Fractions with Common Denominator:

The common denominator for both fractions is [tex]\((x - 5)(x + 3)(x + 7)\)[/tex].

Rewrite each fraction with the common denominator:
[tex]\[ \frac{x}{(x - 5)(x + 3)} = \frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)} \][/tex]

[tex]\[ \frac{4}{(x - 5)(x + 7)} = \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)} \][/tex]

3. Combine the Fractions:

Now calculate the difference of the fractions:
[tex]\[ \frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)} - \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)} \][/tex]

Combine the numerators over the common denominator:
[tex]\[ \frac{x(x + 7) - 4(x + 3)}{(x - 5)(x + 3)(x + 7)} \][/tex]

4. Simplify the Numerator:

Simplify the expression in the numerator:
[tex]\[ x(x + 7) - 4(x + 3) = x^2 + 7x - 4x - 12 = x^2 + 3x - 12 \][/tex]

5. Final Combined Expression:

So the combined fraction simplifies to:
[tex]\[ \frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)} \][/tex]

The final simplified difference between the given fractions is:
[tex]\[ \frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)} \][/tex]

Thus, the correct answer among the options provided is:

[tex]\[ \boxed{\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}} \][/tex]