3. (Partial Fraction Decomposition) a. Write the form of the partial fraction decomposition for: 3 + 2 3 x+ 2 (2 + 3)2(24 + 2.02 + 1) ( r + 3 ) 2 ( 12 + 1 ) 2 b. Write the form of the partial fraction decomposition for: 613 + 4x2 - 5 6 W + 42 - 5 15 - 7x4 + 10x3 123 (12- 5 ) ( x-2)( ) 3 x + 2 = A ( X + 3 ) ( 2( 2 + 1 ) 2 + B ( 2 ( 2 + 1 ) 2 + ( (2C + D ) ( 2C + 3 ) " ( s c 2 + 1 ) - HERE ) (or+3)? put 3 + 3 =0 , we get B = - I equating the coefficient of like powered of x , we get A+ C = O 3 A + B + 6C +D = 0 2 A + 10C + 6D + E= 0 6 A t N B + 6 C + 10 D + 6 E + F = 0 A +9C +GD +9E + 6 F = 3 3A +B +9D +9 F = 2 use matrix method to solve these equations, we get O 1 O O O O 0 0 0 0 3 6 O O O O 3 O 0 0 2 0 10 6 O O 0 08 6 O 6 2 6 10 6 O O 0 O O 6 9 6 100 54 O O O O 0 O 3 O g 6 3 O O 0 O O 100 34 At( = 0 B + 3 C + D = 0 8 C + 6D + E= 0 10 0 D + 54E +8F = 0 & E +OF = 3 100F = 34 Solving these we get A= -54 (= 54 D- - 92 E = 12 [= 34 1000 2 B= - 70 100 100 1000 9 1000 1000 - 54 1 - 70 1 54 x7 - 92 ( 2(+3)2 (2 2 +1 ) 2 1000 ( 2 13) 1000 ( 2( + 3 ) 2 + 7009 Too * + 34 + 100 (X2+1 ] (2 2 71 ) 2