Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 6x - 2 x(x² + 4)2 Submit Answer

Answer 1

Sol? let Betc A + t x zł+ 4 us considero, Dx+E 62-2 x(x+9) (2²+4)~ GX-2 - A() + x(x+ 4) (Bx+c) + x (Dx+ E) i's - -2. z 16 A ie A = 8 put x=0, put x = -1 A 2- -100 -8 = 25A -5(-B+C)-(-D+E) →.-82-25 +53-5C + D - 6 [put - 3 5B-5C + D - E 2 39 8 1

put x=1, 4 = 25A +5(B+C)+(D+E) 9+25 357 2. . → 5B + 50 +D+E = 8 put x 2-2, -19=-8 -16(-2B+C) -2(-2) +E) 323-160 +10-26 3-5 3 16 B - 80 + 2D - E = NG put x=2, 10 2 -8 + 16 (28+ c) + 2 (2D+E) → 323 + 16C + 4D + 2E 2 18 (1) 16B + 8C + 2D + E = 9 Now solve 2 we get, ܝܟ (1) (2 → -10C -252-36 + ② => 10B + 20 = 18 8 ie 5B + D 2 9 ie 5C + E 36 5 6 Now solve 6 & we get, 3-0 = -160-2E =- 23 + + 323 + 4D 2 13 2 23 il 8372 ie 8C + E - 23 9 se + E= ie SB i D = 13 8 7) 8 Again solve 6 & ② we get, B= $ 29 D = , 크 24 B 6 6 & (8 we get, Again solve ez-te 2 E2 77 12

Thees the values are D= 7 24 E = 77 국, 12 Az-; B = 5; c=-ta so, the partial fraction decomposition is 2 + (2244)~ 77 12 + } 12 72 21 6x-2 x.(2244) 8 + + Ⓡx xt 4 62-2 2X -1 + + x(x+9)/ 82 12(x + 1) 7x + 159 24 (x+ ajr