SECTION A.5 Rational Expressions A37 3 Add and Subtract Rational Expressions The rules for adding and subtracting rational expressions are the same as the rules for adding and subtracting rational numbers. If the denominators of two rational expressions to be added (or subtracted) are equal, then add (or subtract) the numerators and keep the common denominator. In Words To add (or subtract) two rational expressions with the same denominator, keep the common denominator and add (or subtract) the numerators. If a b and c d are two rational expressions, then a b c d ac bd b d if 0, 0 ⋅ = ≠ ≠ (2) a b c d a b d c ad bc b c d if 0, 0, 0 = ⋅ = ≠ ≠ ≠ (3) In using equations (2) and (3) with rational expressions, be sure first to factor each polynomial completely so that common factors can be divided out. Leave your answer in factored form. Multiplying and Dividing Rational Expressions Perform the indicated operation and simplify the result. Leave your answer in factored form. (a) − + + ⋅ + + − x x x x x x x 2 1 4 4 2 2 3 2 2 (b) + − − − − x x x x x 3 4 12 8 2 2 3 Solution EXAMPLE 3 (a) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) − + + ⋅ + + − = − + ⋅ + + − = − + + + − = − + ≠ − x x x x x x x x x x x x x x x x x x x x x x x 2 1 4 4 2 1 1 4 1 2 1 1 4 1 1 2 1 4 1 2 2, 0, 1 2 3 2 2 2 2 2 2 2 2 (b) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )( )( ) ( )( ) + − − − − = + − ⋅ − − − = + − + ⋅ − + + − + = + − + + − + − + = + + + − ≠ − − 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 x 3 4 12 8 3 4 8 12 3 2 2 2 2 4 4 3 3 2 2 4 2 2 4 3 2 4 2 4 3, 2, 2, 4 2 2 3 2 3 2 2 2 2 Now Work problems 15 and 21
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