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Chapter 7: Problem 91

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3}{x^{2}+4 x y+3 y^{2}}-\frac{5}{x^{2}-2 x y-3 y^{2}}+\frac{2}{x^{2}-9 y^{2}}$$

Short Answer

Expert verified
The simplified result of the operation is \( -\frac{22y}{(x+3y)(x + y)(x - 3y)}\)

Step by step solution

01

Factor each denominator

The first task is to factor each denominator to allow identification of a common denominator. Factoring gives \( x^{2}+4xy+3y^{2} = (x+3y)(x+y)\), \( x^{2}-2xy-3y^{2} = (x-3y)(x+y)\), and \( x^{2}-9y^{2} = (x-3y)(x+3y)\). So, the given expression becomes: \(\frac{3}{(x+3y)(x+y)}\) - \(\frac{5}{(x-3y)(x+y)}\) + \(\frac{2}{(x-3y)(x+3y)}\)
02

Find a common denominator

Looking at the factored denominators, a common denominator amongst the three terms will be \((x + 3y)(x + y)(x - 3y)\). Each of the expressions has to be multiplied by the term it lacks to get to this denominator.
03

Rewrite the expression with the common denominator

Rewrite each term of the expression with the common denominator: \(\frac{3(x - 3y)}{(x+3y)(x + y)(x - 3y)}\) - \(\frac{5(x + 3y)}{(x+3y)(x - y)(x - 3y)}\) + \(\frac{2(x + y)}{(x - 3y)(x + 3y)(x + y)}\)
04

Perform the operation and simplify

When the terms are multiplied out the expression simplifies to : \(\frac{3x - 9y - 5x - 15y + 2x + 2y}{(x+3y)(x + y)(x - 3y)}\). Further simplification leads to: \(\frac{3x - 5x + 2x - 9y - 15y + 2y}{(x+3y)(x + y)(x - 3y)}\) = \(\frac{0x - 22y}{(x+3y)(x + y)(x - 3y)}\) = \( -\frac{22y}{(x+3y)(x + y)(x - 3y)}\)

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