A40 APPENDIX Review Using the Least Common Multiple to Add Rational Expressions Perform the indicated operation and simplify the result. Leave your answer in factored form. + + + − − ≠ − − x x x x x x 3 2 2 3 1 2, 1, 1 2 2 Solution EXAMPLE 7 Step 1 Factor completely the polynomials in the denominators. ( )( ) ( )( ) + + = + + − = − + x x x x x x x 3 2 2 1 1 1 1 2 2 Step 2 The LCM is ( )( )( ) + + − x x x 2 1 1 . Do you see why? Step 3 Write each rational expression using the LCM as the denominator. ( )( ) ( )( ) ( ) ( )( )( ) + + = + + = + + ⋅ − − = − + + − x x x x x x x x x x x x x x x x 3 2 2 1 2 1 1 1 1 2 1 1 2 ( )( ) ( )( ) ( )( ) ( )( )( ) − − = − − + = − − + ⋅ + + = − + − + + x x x x x x x x x x x x x x x 2 3 1 2 3 1 1 2 3 1 1 2 2 2 3 2 1 1 2 2 Multiply numerator and denominator by −x 1 to get the LCM in the denominator. æ Multiply numerator and denominator by +x 2 to get the LCM in the denominator. æ Step 4 Now add by using equation (4). ( ) ( ) ( ) ( ) ( )( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( ) + + + − − = − + + − + − + + + − = − + + − + + − = − + + − = − + + − x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 3 2 2 3 1 1 2 1 1 2 3 2 2 1 1 2 6 2 1 1 3 6 2 1 1 3 2 2 1 1 2 2 2 2 2 2 Now Work problem 29 5 Simplify Complex Rational Expressions When sums and/or differences of rational expressions appear as the numerator and/ or denominator of a quotient, the quotient is called a complex rational expression.* For example, + − − − − + − x x x x x x 1 1 1 1 and 4 3 3 2 1 2 2 are complex rational expressions. To simplify a complex rational expression means to write it as a rational expression reduced to lowest terms. This can be accomplished in either of two ways. *Some texts use the term complex fraction.
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