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Chapter 7: Problem 86

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3}{x^{2}-49}+\frac{2}{x^{2}-15 x+56}-\frac{5}{x^{2}-x-56}$$

Short Answer

Expert verified
The simplified form of the given expression is \[\frac{25}{(x-7)(x+7)(x-8)}\].

Step by step solution

01

Factorize the Denominators

To find a common denominator, first factorize the denominators of each fraction. The denominators are: \(x^{2}-49\), \(x^{2}-15x+56\), and \(x^{2}-x-56\). These can be factorized as follows: \(x^{2}-49 = (x-7)(x+7)\), \(x^{2}-15x+56 = (x-7)(x-8)\), and \(x^{2}-x-56 = (x-8)(x+7)\).
02

Find the Common Denominator

The common denominator of the fractions will be the product of all the distinct factors from previous factored denominators, which is \((x-7)(x+7)(x-8)\).
03

Rewrite the Fractions with the Common Denominator

Express each fraction with this new denominator and simplify the numerators. The fractions can now be rewritten as follows: \[\frac{3}{(x-7)(x+7)} * \frac{(x-8)}{(x-8)} + \frac{2}{(x-7)(x-8)} * \frac{(x+7)}{(x+7)} - \frac{5}{(x-8)(x+7)} * \frac{(x-7)}{(x-7)}\] \n Simplifying this gives the new fractions as: \[\frac{3(x-8)}{(x-7)(x+7)(x-8)} + \frac{2(x+7)}{(x-7)(x+7)(x-8)} - \frac{5(x-7)}{(x-8)(x+7)(x-7)}\].
04

Simplify the Fractions and Combined Them

Cancelling out similar terms in the numerators and denominators of the fractions, we get: \[\frac{3x-24}{(x-7)(x+7)(x-8)} + \frac{2x+14}{(x-7)(x+7)(x-8)} - \frac{5x-35}{(x-7)(x+7)(x-8)}\]. \n Now that the fractions have the same denominators we can add or subtract the numerators: \[\frac{3x-24 + 2x+14 - 5x+35}{(x-7)(x+7)(x-8)}\]. \n Simplifying the numerator we get: \[\frac{0x+25}{(x-7)(x+7)(x-8)}\]. \n This simplifies to: \[\frac{25}{(x-7)(x+7)(x-8)}\].

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