A36 APPENDIX Review Cancellation Property = ≠ ≠ ac bc a b b c if 0, 0 (1) CAUTION Use the Cancellation Property only with rational expressions written in factored form. Be sure to divide out only common factors, not common terms! j Reducing a Rational Expression to Lowest Terms Reduce to lowest terms: + + + + x x x x 4 4 3 2 2 2 Solution EXAMPLE 1 Begin by factoring the numerator and the denominator. ( )( ) ( )( ) + + = + + + + = + + x x x x x x x x 4 4 2 2 3 2 2 1 2 2 Since a common factor, +x 2, appears, the original expression is not in lowest terms. To reduce it to lowest terms, use the Cancellation Property: ( )( ) ( )( ) + + + + = + + + + = + + ≠ − − x x x x x x x x x x x 4 4 3 2 2 2 2 1 2 1 2, 1 2 2 2 Multiply and Divide Rational Expressions The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers. Rational expressions are described in the same manner as rational numbers. In expression (a), the polynomial + x 1 3 is the numerator , and x is the denominator . When the numerator and denominator of a rational expression contain no common factors (except 1 and −1), we say that the rational expression is reduced to lowest terms , or simplified . The polynomial in the denominator of a rational expression cannot be equal to 0 because division by 0 is not defined. For example, for the expression + x x 1 , 3 x cannot take on the value 0. The domain of the variable x is { } ≠ x x 0 . A rational expression is reduced to lowest terms by factoring the numerator and the denominator completely and dividing out any common factors using the Cancellation Property: Reducing Rational Expressions to Lowest Terms Reduce each rational expression to lowest terms. (a) − − x x x 8 2 3 3 2 (b) − − − x x x 8 2 12 2 Solution EXAMPLE 2 (a) ( ) ( ) ( ) − − = − + + − = + + ≠ x x x x x x x x x x x x 8 2 2 2 4 2 2 4 0, 2 3 3 2 2 2 2 2 (b) ( ) ( )( ) ( )( ) ( )( ) − − − = − − + = − − − + = − + ≠− x x x x x x x x x x x 8 2 12 2 4 4 3 2 1 4 4 3 2 3 3, 4 2 Now Work problem 7
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