Section 7. Partial Fractions 0. 0 7 0 0 0 0 Solution:, 0 Equation Equation Eq. Eq. 07. nswers will var. Section 7. Partial Fractions N You should know how to decompose a rational function into partial fractions. D (a) If the fraction is improper, divide to obtain N D p N D (a) where p is a polnomial. (b) Factor the denominator completel into linear and irreducible quadratic factors. (c) For each factor of the form p q m, the partial fraction decomposition includes the terms p q p q... m p q. m (d) For each factor of the form a b c n, the partial fraction decomposition includes the terms a b c a b c... n n a b c. n You should know how to determine the values of the constants in the numerators. N (a) Set partial fraction decomposition. D (b) Multipl both sides b D to obtain the basic equation. (c) For distinct linear factors, substitute the zeros of the distinct linear factors into the basic equation. (d) For repeated linear factors, use the coefficients found in part (c) to rewrite the basic equation. Then use other values of to solve for the remaining coefficients. (e) For quadratic factors, epand the basic equation, collect like terms, and then equate the coefficients of like terms. Vocabular heck. partial fraction decomposition. improper. m; n; irreducible. basic equation. Matches (b).. Matches (c).. Matches (d).
hapter 7 Sstems of Equations and Inequalities. Matches (a).. 7 7. 7. 0 0 0. 9. 0. D. 0 0 0.. D E. D. Let : Let :. 9 Let : Let : 9 7. Let 0: Let :. Let : Let 0: 9. Let : Let 0:
Section 7. Partial Fractions 0. Let : Let :. Let : Let :.,. Let 0: Let : Let : 0. Let 0: Let :. Let 0: Let : Let :. Let : Let 0: 7. Let : 9 Let 0: 0 0 9 9
hapter 7 Sstems of Equations and Inequalities. D Let 0 : Let : 7 D Substitute and D into the equation, epand the binomials, collect like terms, and equate the coefficients of like terms. or 0 7 D 9. Equating coefficients of like terms gives, 0, and. Therefore,,, and 0. 0. Equating coefficients of like powers gives 0,, and 0. Substituting for and for in the second equation gives, so and,,.. Equating coefficients of like terms gives 0,, and 0. Therefore,,, and.. Let : 9 Let : 0 Let : 9 9 0 9
Section 7. Partial Fractions. Equating coefficients of like terms gives 0, D, 0, and 0 D. Using the first and third equation, we have 0 and 0; b subtraction, 0. Using the second and fourth equation, we have D and D 0; b subtraction, D, so D. Substituting 0 for and for D in the first and second equations, we have 0 and so and,. D D D D D D D. Equating coefficients of like powers: 0 D D D D 0 D. D ONTINUED D D D D D D
hapter 7 Sstems of Equations and Inequalities. ONTINUED Equating coefficients of like terms gives 0, 0 D,, and 0 D. Using the first and third equations, we have 0 and ; b subtraction,, so. Using the second and fourth equations, we have D 0 and D 0; b subtraction D 0, so D 0. Substituting for and 0 for D in the first and second equations, we have and 0, so and.. Equating coefficients of like powers gives Therefore,,, and. 0,, and. 7. Equating coefficients of like terms gives, 0, and. Subtracting both sides of the second equation from the first gives ; combining this with the third equation gives and. Since, we also have 0.. 7 7 Equating coefficients of like terms gives,, and 7. dding the second and third equations, and subtracting the first, gives, so. Therefore,,, and. 7
Section 7. Partial Fractions 7 9. 0. Using long division gives.. 7 9 9 9 Let : Let : 7 7 7 7. Using long division gives: 0 7 7 7
hapter 7 Sstems of Equations and Inequalities. Let : Let 0: Let : So, and. 0. Let : Equating coefficients of like powers:,. Let : 9 Let :. 7 7 Let 0: Let : Let : 7
Section 7. Partial Fractions 9 7. Let : Let 0: Let : 0 0 0. Let 0: Let : Let : 0 9. D D Equating coefficients of like powers: 0 D D 0 D 0. Let : D D Let : D D ONTINUED
70 hapter 7 Sstems of Equations and Inequalities 0. ONTINUED Equating coefficients of like powers: 0. Let : Let : 9. Let : Let : 0 9 9 7 0. (a) (b) ( Let 0: Let : 0 0 0 Vertical asmptotes: 0 Vertical asmptote: 0 Vertical asmptote: and (c) The combination of the vertical asmptotes of the terms of the decomposition are the same as the vertical asmptotes of the rational function.
Section 7. Partial Fractions 7. (a) Equating coefficients of like powers gives,, and. Therefore,, 0, and. (b) Vertical asmptote at 0 and = + = has vertical asmptote 0. (c) The vertical asmptote of is the same as the vertical asmptote of the rational function.. (a) 9 ( Let : Let : 0 9 (b) 9 Vertical asmptotes: ± Vertical asmptote: Vertical asmptote: (c) The combination of the vertical asmptotes of the terms of the decomposition are the same as the vertical asmptotes of the rational function.. (a) 9 0 D 0 9 0 0 D 0 7 0 0 D Equating coefficients of like powers gives 0, 0 D, 0 0, and 7. Since 7,. Therefore, 0,, 0, and D. 9 0 0 0 D 0 ONTINUED
7 hapter 7 Sstems of Equations and Inequalities. ONTINUED (b) 9 0 and 0 0 0 = = 0 + 0 0 Vertical asmptote is 0. has vertical asmptote 0. (c) The vertical asmptote of is the same as the vertical asmptote of the rational function. 7. (a) 000 77 7 7, 0 < Let Let 000 7 7 0,000 : 7 7 7 000 7 : 00 000 000 000 000 000 000 77 7 7 7 7, 0 < (b) Y ma 000 (c) 7 000 Yma Y min 7 Ymin 000 (d) 0 00 Y ma 0. 00F Y min 0..7F. One wa to find the constants is to choose values of the variable that eliminate one or more of the constants in the basic equation so that ou can solve for another constant. If necessar, ou can then use these constants with other chosen values of the variable to solve for an remaining constants. nother wa is to epand the basic equation and collect like terms. Then ou can equate coefficients of the like terms on each side of the equation to obtain simple equations involving the constants. If necessar, ou can solve these equations using substitution. 9. False. The partial fraction decomposition is 0 0 0. 0. False. The epression is an improper rational epression, so ou must first divide before appling partial fraction decomposition.. a a a a a Let a: a a Let a: a a a a a a, a is a constant.. a is a constant. a a, a Let 0: a a Let a: a a a a a
Section 7. Partial Fractions 7. a a a Let 0: a a Let a: a a a a a. a is a positive integer. a a, a Let : a a Let a: a a a a a. f 9. f 9 Intercepts: 9 0,,, 0,, 0 Verte: 9 Graph rises to the left and, rises to the right. -intercepts:, 0,, 0 0 0 0 7. f. f Intercepts: 0, 0,, 0 Graph rises to the left and falls to the right. Intercepts: 0,,, 0 9. f -intercepts:, 0,, 0 -intercept: 0, Vertical asmptote: Slant asmptote: No horizontal asmptote. 0 0 0 0 70. f -intercept: Vertical asmptotes: Horizontal asmptote:, 0 0 and 0