SECTION A.5 Rational Expressions A41 Both options are shown in the next example. By carefully studying each option, you can discover situations in which one may be easier to use than the other. Simplifying a Complex Rational Expression Option 1 Treat the numerator and denominator of the complex rational expression separately, performing whatever operations are indicated and simplifying the results. Follow this by simplifying the resulting rational expression. O ption 2 Find the LCM of the denominators of all rational expressions that appear in the complex rational expression. Multiply the numerator and denominator of the complex rational expression by the LCM and simplify the result. Multiply the numerator and denominator by x4 . æ Use the Distributive Property in the numerator. æ Simplify. æ Factor. æ Rule for multiplying quotients æ Rule for dividing quotients æ Rule for adding quotients æ Simplifying a Complex Rational Expression Simplify: + + ≠ − x x x 1 2 3 3 4 3, 0 Solution EXAMPLE 8 Option 1 First, perform the indicated operation in the numerator, and then divide. x x x x x x x x x x x x x x x x x x x x 1 2 3 3 4 1 2 3 2 3 4 6 2 3 4 6 2 4 3 6 4 2 3 2 2 6 2 3 2 6 3 ( ) ( ) ( ) ( ) ( ) ( ) + + = ⋅ + ⋅ ⋅ + = + + = + ⋅ + = + ⋅ ⋅ ⋅ + = ⋅ ⋅ + ⋅ ⋅ + = + + Option 2 The rational expressions that appear in the complex rational expression are + x x 1 2 , 3 , 3 4 The LCM of their denominators is x4 . Multiply the numerator and denominator of the complex rational expression by x4 and then simplify. x x x x x x x x x x x x x x x x x x x x x x 1 2 3 3 4 4 1 2 3 4 3 4 4 1 2 4 3 4 3 4 2 2 1 2 4 3 4 3 4 2 12 3 2 6 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = ⋅ + ⋅ + = ⋅ + ⋅ ⋅ + = ⋅ ⋅ + ⋅ ⋅ + = + + = + +
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