Problem 91 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \(\frac{3}{(x+3y)(x+y)}\) - \(\frac{5}{(x-3y)(x+y)}\) + \(\frac{2}{(x-3y)(x+3y)}\)

Step 2: 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.

Step 3: Rewrite the expression with the common denominator

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

Step 4: 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: \(\frac{3x - 5x + 2x - 9y - 15y + 2y}{(x+3y)(x + y)(x - 3y)}\) = \(\frac{0x - 22y}{(x+3y)(x + y)(x - 3y)}\) = \( -\frac{22y}{(x+3y)(x + y)(x - 3y)}\)