77 R.7 Rational Expressions y1-12 1y - 221-12 + 8 2 - y Multiply the first expression by -1 in both the numerator and the denominator. = -y 2 - y + 8 2 - y Simplify. = 8 - y 2 - y (c) 3 x2 + x - 2 - 1 x2 - x - 12 = 3 1x - 121x + 22 - 1 1x + 321x - 42 Factor. The LCD is 1x - 121x + 221x + 321x - 42. = 31x + 321x - 42 1x - 121x + 221x + 321x - 42 - 11x - 121x + 22 1x + 321x - 421x - 121x + 22 = 31x2 - x - 122 - 1x2 + x - 22 1x - 121x + 221x + 321x - 42 Multiply in the numerators, and then subtract them. = 3x2 - 3x - 36 - x2 - x + 2 1x - 121x + 221x + 321x - 42 Distributive property = 2x2 - 4x - 34 1x - 121x + 221x + 321x - 42 Combine like terms in the numerator. S Now Try Exercises 55, 63, and 69. Be careful with signs. This equivalent expression results. CAUTION When subtracting fractions as in Example 4(c), be sure to distribute the negative sign to each term in the numerator. Complex Fractions The quotient of two rational expressions is a complex fraction. There are two methods for simplifying a complex fraction. EXAMPLE 5 Simplifying Complex Fractions Simplify each complex fraction. (a) 6 - 5 k 1 + 5 k (b) a a + 1 + 1 a 1 a + 1 a + 1 SOLUTION (a) Method 1 for simplifying uses the identity property for multiplication. We multiply both numerator and denominator by the LCD of all the fractions, k. 6 - 5 k 1 + 5 k = ka6 - 5 kb ka1 + 5 kb = 6k - ka 5 kb k + ka 5 kb = 6k - 5 k + 5 Distribute k to all terms within the parentheses.
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