75 R.7 Rational Expressions (c) 3p2 + 11p - 4 24p3 - 8p2 , 9p + 36 24p4 - 36p3 = 1p + 4213p - 12 8p213p - 12 , 91p + 42 12p312p - 32 Factor. = 1p + 4213p - 12 8p213p - 12 # 12p312p - 32 91p + 42 Multiply by the reciprocal of the divisor. = 12p312p - 32 9 # 8p2 Divide out common factors. Multiply fractions. = 3 # 4 # p2 # p12p - 32 3 # 3 # 4 # 2 # p2 Factor. = p12p - 32 6 Lowest terms (d) x3 - y3 x2 - y2 # 2x + 2y + xz + yz 2x2 + 2y2 + zx2 + zy2 = 1x - y21x2 + xy + y22 1x + y21x - y2 # 21x + y2 + z1x + y2 21x2 + y22 + z1x2 + y22 Factor. Group terms and factor. = 1x - y21x2 + xy + y22 1x + y21x - y2 # 1x + y212 + z2 1x2 + y2212 + z2 Factor by grouping. = x2 + xy + y2 x2 + y2 Divide out common factors. Multiply fractions. S Now Try Exercises 33, 43, and 47. Addition and Subtraction We add and subtract rational expressions in the same way that we add and subtract fractions. Adding and Subtracting Rational Expressions For fractions a b and c d 1b≠0, d≠02, the following hold true. a b + c d = ad +bc bd and a b − c d = ad −bc bd To add (or subtract) two fractions in practice, find their least common denominator (LCD) and change each fraction to one with the LCD as denominator. The sum (or difference) of their numerators is the numerator of their sum (or difference), and the LCD is the denominator of their sum (or difference). Finding the Least Common Denominator (LCD) Step 1 Write each denominator as a product of prime factors. Step 2 Form a product of all the different prime factors. Each factor should have as exponent the greatest exponent that appears on that factor.
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