SECTION A.5 Rational Expressions A39 ( ) 5b æ (b) x x x x x x x x x x x x x x x x x x x x 4 1 4 4 4 1 4 1 4 4 2 2 2 2 2 2 2 2 2 2 2 3 2 ( ) ( ) ( ) ( ) ( )( ) − − = − ⋅ − − − ⋅ = − − − ⋅ = − + − + Now Work problem 25 4 Use the Least Common Multiple Method If the denominators of two rational expressions to be added (or subtracted) have common factors, we usually do not use the general rules given by equations (5a) and (5b). Just as with fractions, we use the least common multiple (LCM) method . The LCM method uses the polynomial of least degree that has each denominator polynomial as a factor. We begin with an example that requires only Steps 1 and 2. The LCM Method for Adding or Subtracting Rational Expressions The Least Common Multiple (LCM) Method requires four steps: Step 1 Factor completely the polynomial in the denominator of each rational expression. Step 2 The LCM of the denominators is the product of each unique factor, with each of these factors raised to a power equal to the greatest number of times that the factor occurs in any denominator. Step 3 Write each rational expression using the LCM as the common denominator. Step 4 Add or subtract the rational expressions using equation (4). Finding the Least Common Multiple Find the least common multiple of the following pair of polynomials: ( ) ( ) ( )( ) − + − + x x x x x 1 1 and 4 1 1 2 3 Solution EXAMPLE 6 Step 1 The polynomials are already factored completely as ( ) ( ) ( )( ) − + − + x x x x x 1 1 and 4 1 1 2 3 Step 2 Start by writing the factors of the left-hand polynomial. (Or you could start with the one on the right.) ( ) ( ) − + x x x 1 1 2 Now look at the right-hand polynomial. Its first factor, 4, does not appear in our list, so we insert it. ( ) ( ) − + x x x 4 1 1 2 The next factor, −x 1, is already in our list, so no change is necessary. The final factor is ( ) +x 1 . 3 Since our list has +x 1 to the first power only, we replace +x 1 in the list by ( ) +x 1 . 3 The LCM is ( ) ( ) − + x x x 4 1 1 2 3 Notice that the LCM is, in fact, the polynomial of least degree that contains ( ) ( ) − + x x x 1 1 2 and ( )( ) − + x x 4 1 1 3 as factors.
RkJQdWJsaXNoZXIy NjM5ODQ=