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0 0 1 133.44 714.00 Tm <706F737369626C6520> Tj 1 0 0 1 176.40 713.28 Tm <746F20> Tj 1 0 0 1 189.12 713.28 Tm <73706C697420> Tj 1 0 0 1 212.88 713.28 Tm <6120> Tj 1 0 0 1 221.28 713.28 Tm <636F6D706C69636174656420> Tj 1 0 0 1 282.72 713.04 Tm <726174696F6E616C20> Tj 1 0 0 1 322.80 712.80 Tm <65787072657373696F6E20> Tj 1 0 0 1 377.52 712.80 Tm <696E746F20> Tj 1 0 0 1 399.84 712.80 Tm <6120> Tj 1 0 0 1 408.24 712.80 Tm <73756D20> Tj 1 0 0 1 431.28 712.80 Tm <6F6620> Tj 1 0 0 1 444.48 712.80 Tm <736D616C6C65722C20> Tj 1 0 0 1 486.00 712.56 Tm <73696D706C6572> Tj 0.24 Tc 1 0 0 1 50.40 699.84 Tm <65787072657373696F6E732E20> Tj 1 0 0 1 112.56 699.60 Tm <5468697320> Tj 1 0 0 1 136.80 699.60 Tm <73756D20> Tj 1 0 0 1 159.84 699.60 Tm <697320> Tj 1 0 0 1 170.88 699.60 Tm <63616C6C656420> Tj 1 0 0 1 203.28 699.36 Tm <74686520> Tj 1 0 0 1 218.64 699.84 Tm <7061727469616C6672616374696F6E20> Tj 1 0 0 1 297.12 699.12 Tm <6465636F6D706F736974696F6E20> Tj 1 0 0 1 370.80 699.12 Tm <6F6620> Tj 1 0 0 1 384.48 699.12 Tm <74686520> Tj 1 0 0 1 401.76 698.88 Tm <6F726967696E616C20> Tj 1 0 0 1 442.08 699.12 Tm <726174696F6E616C20> Tj 1 0 0 1 482.64 698.88 Tm <65787072657373696F6E2E> Tj 0.24 Tc 1 0 0 1 50.64 685.68 Tm <53657474696E6720> Tj 1 0 0 1 87.12 685.92 Tm <757020> Tj 1 0 0 1 102.24 685.92 Tm <74686520> Tj 1 0 0 1 119.76 685.92 Tm <6465636F6D706F736974696F6E20> Tj 1 0 0 1 194.16 685.68 Tm <646570656E647320> Tj 1 0 0 1 236.16 685.44 Tm <6F6E20> Tj 1 0 0 1 251.04 685.20 Tm <74686520> Tj 1 0 0 1 269.28 685.20 Tm <666163746F727320> Tj 1 0 0 1 305.04 684.96 Tm <6F6620> Tj 1 0 0 1 317.76 685.20 Tm <74686520> Tj 1 0 0 1 335.76 685.20 Tm <64656E6F6D696E61746F722C20> Tj 1 0 0 1 402.48 684.96 Tm <6275742E74686520> Tj 1 0 0 1 438.96 685.44 Tm <70726F6365737320> Tj 1 0 0 1 478.56 684.96 Tm <6F6620> Tj 1 0 0 1 492.00 684.96 Tm <736F6C76696E6720> Tj 1 0 0 1 529.68 684.72 Tm <746865> Tj 0.24 Tc 1 0 0 1 50.16 672.00 Tm <6465636F6D706F736974696F6E20> Tj 1 0 0 1 123.84 672.00 Tm <697320> Tj 1 0 0 1 134.64 671.76 Tm <74686520> Tj 1 0 0 1 152.88 671.76 Tm <73616D652E> Tj 0.24 Tc 1 0 0 1 49.92 644.40 Tm <496E20> Tj 1 0 0 1 63.12 644.64 Tm <67656E6572616C2C20> Tj 1 0 0 1 104.40 644.16 Tm <74686520> Tj 1 0 0 1 121.92 644.16 Tm <6465636F6D706F736974696F6E20> Tj 1 0 0 1 195.84 644.16 Tm <77696C6C20> Tj 1 0 0 1 217.68 643.92 Tm <6861766520> Tj 1 0 0 1 243.36 643.68 Tm <6F6E6520> Tj 1 0 0 1 264.00 643.68 Tm <6672616374696F6E20> Tj 1 0 0 1 304.56 643.44 Tm <666F7220> Tj 1 0 0 1 321.36 643.44 Tm <6561636820> Tj 1 0 0 1 346.56 643.44 Tm <666163746F7220> Tj 1 0 0 1 377.76 643.44 Tm <696E20> Tj 1 0 0 1 390.00 643.44 Tm <74686520> Tj 1 0 0 1 408.24 643.44 Tm <64656E6F6D696E61746F722E20> Tj 1 0 0 1 475.44 643.20 Tm <496620> Tj 1 0 0 1 486.24 643.20 Tm <74686520> Tj 1 0 0 1 504.48 643.20 Tm <666163746F7220> Tj 1 0 0 1 535.20 643.20 Tm <6973> Tj 0.24 Tc 1 0 0 1 49.92 630.48 Tm <6C696E6561722C20> Tj 1 0 0 1 84.48 630.48 Tm <74686520> Tj 1 0 0 1 102.24 630.24 Tm <6E756D657261746F7220> Tj 1 0 0 1 154.56 630.24 Tm <6F6620> Tj 1 0 0 1 167.52 630.24 Tm <74686520> Tj 1 0 0 1 185.76 630.00 Tm <6672616374696F6E20> Tj 1 0 0 1 225.84 630.00 Tm <697320> Tj 1 0 0 1 235.68 631.44 Tm <6A75737420> Tj 1 0 0 1 257.52 630.00 Tm <616E20> Tj 1 0 0 1 271.44 629.76 Tm <756E6B6E6F776E20> Tj 1 0 0 1 319.68 629.76 Tm <636F6E7374616E742C20> Tj 1 0 0 1 366.00 629.76 Tm <6C696B6520> Tj 1 0 0 1 387.12 629.52 Tm <412E20> Tj 1 0 0 1 405.12 629.52 Tm <496620> Tj 1 0 0 1 416.16 629.52 Tm <74686520> Tj 1 0 0 1 434.16 629.52 Tm <666163746F7220> Tj 1 0 0 1 465.12 629.52 Tm <697320> Tj 1 0 0 1 476.16 629.76 Tm <717561647261637469632C746865> Tj 0.24 Tc 1 0 0 1 49.68 616.56 Tm <6E756D657261746F7220> Tj 1 0 0 1 102.00 616.56 Tm <77696C6C20> Tj 1 0 0 1 123.60 616.56 Tm <6861766520> Tj 1 0 0 1 149.76 616.56 Tm <6120> Tj 1 0 0 1 158.16 616.32 Tm <6C696E65617220> Tj 1 0 0 1 188.64 616.32 Tm <666F726D2C20> Tj 1 0 0 1 218.16 616.08 Tm <6C696B6520> Tj 1 0 0 1 238.80 616.08 Tm <4122722B422E> Tj 0.24 Tc 1 0 0 1 49.68 588.48 Tm <546F20> Tj 1 0 0 1 66.24 588.48 Tm <736F6C766520> Tj 1 0 0 1 94.80 588.48 Tm <6120> Tj 1 0 0 1 102.96 588.48 Tm <6465636F6D706F736974696F6E2C20> Tj 1 0 0 1 179.52 588.24 Tm <776520> Tj 1 0 0 1 196.80 588.24 Tm <736F6C766520> Tj 1 0 0 1 225.12 588.00 Tm <666F7220> Tj 1 0 0 1 241.92 588.00 Tm <74686520> Tj 1 0 0 1 259.44 588.00 Tm <756E6B6E6F776E20> Tj 1 0 0 1 307.68 587.76 Tm <6E756D657261746F72732E20> Tj 1 0 0 1 371.28 587.76 Tm <4C6574277320> Tj 1 0 0 1 399.36 587.76 Tm <6C6F6F6B20> Tj 1 0 0 1 424.08 587.52 Tm <617420> Tj 1 0 0 1 435.84 587.52 Tm <6120> Tj 1 0 0 1 444.24 587.52 Tm <73696D706C6520> Tj 1 0 0 1 479.52 587.52 Tm <6578616D706C652E> Tj 0.24 Tc 1 0 0 1 85.68 552.24 Tm <45782E20> Tj 1 0 0 1 105.12 552.24 Tm <46696E6420> Tj 1 0 0 1 129.84 552.24 Tm <74686520> Tj 1 0 0 1 147.36 552.72 Tm <7061727469616C20> Tj 1 0 0 1 181.92 552.00 Tm <6672616374696F6E20> Tj 1 0 0 1 222.00 551.76 Tm <6465636F6D706F736974696F6E20> Tj 1 0 0 1 296.16 551.52 Tm <6F6620> Tj 1 0 0 1 324.96 548.40 Tm <2E20> Tj 1 0 0 1 341.28 559.20 Tm <2A49296F20> Tj 1 0 0 1 377.76 548.40 Tm <5E2C20> Tj 1 0 0 1 391.20 551.52 Tm <2E> Tj 1.44 Tc 1 0 0 1 253.68 549.36 Tm <5E20> Tj 1 0 0 1 324.00 542.64 Tm <28782D342928782B3229> Tj 0.24 Tc 1 0 0 1 49.92 512.64 Tm <5374657020> Tj 1 0 0 1 74.88 512.88 Tm <6C3A20> Tj 1 0 0 1 89.52 512.64 Tm <53657420> Tj 1 0 0 1 107.76 512.88 Tm <757020> Tj 1 0 0 1 122.40 512.64 Tm <74686520> Tj 1 0 0 1 139.92 513.12 Tm <7061727469616C20> Tj 1 0 0 1 174.24 512.64 Tm <6672616374696F6E20> Tj 1 0 0 1 214.56 512.40 Tm <6465636F6D706F736974696F6E20> Tj 1 0 0 1 288.48 512.16 Tm <7769746820> Tj 1 0 0 1 312.96 512.16 Tm <74686520> Tj 1 0 0 1 330.72 511.92 Tm <756E6B6E6F776E20> Tj 1 0 0 1 378.72 511.92 Tm <636F6E7374616E747320> Tj 1 0 0 1 426.96 511.92 Tm <412C20> Tj 1 0 0 1 441.60 511.92 Tm <422C20> Tj 1 0 0 1 456.00 511.44 Tm <432C20> Tj 1 0 0 1 470.16 511.68 Tm <6574632C20> Tj 1 0 0 1 490.08 511.68 Tm <696E20> Tj 1 0 0 1 502.32 511.68 Tm <746865> Tj 0.24 Tc 1 0 0 1 49.20 498.72 Tm <6E756D657261746F7220> Tj 1 0 0 1 102.00 498.72 Tm <6F6620> Tj 1 0 0 1 114.72 498.72 Tm <74686520> Tj 1 0 0 1 132.24 498.72 Tm <6465636F6D706F736974696F6E2E> Tj 1.50 Tc 1 0 0 1 169.20 476.16 Tm <2C272C20> Tj 1 0 0 1 197.76 483.84 Tm <2A7421296120> Tj 1 0 0 1 233.76 473.04 Tm <3D2C3D20> Tj 1 0 0 1 266.40 483.60 Tm <412E2A20> Tj 1 0 0 1 303.12 483.36 Tm <423D2C616E646F7572676F616C6973746F66696E6441616E64422E> Tj 1.50 Tc 1 0 0 1 180.48 466.80 Tm <28782D342928782B322920> Tj 1 0 0 1 258.48 466.08 Tm <782D3420> Tj 1 0 0 1 295.20 466.08 Tm <782B3227> Tj 0.24 Tc 1 0 0 1 49.68 450.00 Tm <5374657020> Tj 1 0 0 1 73.44 450.24 Tm <323A20> Tj 1 0 0 1 89.04 450.00 Tm <4D756C7469706C7920> Tj 1 0 0 1 133.68 450.00 Tm <626F746820> Tj 1 0 0 1 158.64 449.76 Tm <736964657320> Tj 1 0 0 1 185.52 449.76 Tm <6F6620> Tj 1 0 0 1 198.48 449.76 Tm <74686520> Tj 1 0 0 1 216.24 449.76 Tm <726573756C74696E6720> Tj 1 0 0 1 261.36 449.52 Tm <6571756174696F6E20> Tj 1 0 0 1 305.76 449.28 Tm <627920> Tj 1 0 0 1 320.64 449.28 Tm <74686520> Tj 1 0 0 1 338.88 449.28 Tm <6C6561737420> Tj 1 0 0 1 364.08 449.28 Tm <636F6D6D6F6E20> Tj 1 0 0 1 409.68 449.28 Tm <64656E6F6D696E61746F722E> Tj 1.50 Tc /F3 20.0 Tf 1 0 0 1 161.52 427.92 Tm <2E272E28782D20> Tj 1 0 0 1 193.20 427.68 Tm <342928782B7A7420> Tj 1 0 0 1 237.12 415.68 Tm <556666693D28782D20> Tj 1 0 0 1 336.24 427.20 Tm <342928782B7A782A2A3474> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 49.44 402.00 Tm <5374657020> Tj 1 0 0 1 73.68 402.00 Tm <333A20> Tj 1 0 0 1 89.28 401.76 Tm <53696D706C69667920> Tj 1 0 0 1 133.44 401.76 Tm <626F746820> Tj 1 0 0 1 158.40 401.76 Tm <736964657320> Tj 1 0 0 1 185.28 401.52 Tm <6F6620> Tj 1 0 0 1 198.24 401.76 Tm <74686520> Tj 1 0 0 1 216.24 401.52 Tm <6571756174696F6E2E20> Tj 1 0 0 1 263.52 401.28 Tm <457665727920> Tj 1 0 0 1 295.44 401.28 Tm <64656E6F6D696E61746F7220> Tj 1 0 0 1 359.52 401.28 Tm <73686F756C6420> Tj 1 0 0 1 395.04 401.04 Tm <63616E63656C20> Tj 1 0 0 1 428.88 401.04 Tm <6F75742E> Tj 0.24 Tc /F3 19.0 Tf 1 0 0 1 50.88 379.92 Tm <3A2E20> Tj 1 0 0 1 60.72 377.52 Tm <40323429402E642920> Tj 1 0 0 1 124.80 367.92 Tm <646820> Tj 1 0 0 1 193.92 381.12 Tm <3D20> Tj 1 0 0 1 203.76 379.92 Tm <28722C2D3429287820> Tj 1 0 0 1 247.68 379.44 Tm <2E20> Tj 1 0 0 1 256.80 379.20 Tm <722920> Tj 1 0 0 1 267.84 367.20 Tm <412B20> Tj 1 0 0 1 313.44 379.44 Tm <287820> Tj 1 0 0 1 325.92 381.84 Tm <2D20> Tj 1 0 0 1 334.80 379.20 Tm <34294F20> Tj 1 0 0 1 355.44 379.20 Tm <46742920> Tj 1 0 0 1 377.76 367.20 Tm <26> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 233.52 353.28 Tm <3A2E20> Tj 1 0 0 1 243.60 353.04 Tm <782B313420> Tj 1 0 0 1 275.28 356.88 Tm <2D20> Tj 1 0 0 1 285.12 353.52 Tm <28782B20> Tj 1 0 0 1 306.96 353.04 Tm <3229412B402D6742> Tj 0.24 Tc 1 0 0 1 49.20 337.44 Tm <5374657020> Tj 1 0 0 1 73.20 337.68 Tm <343A20> Tj 1 0 0 1 88.80 337.44 Tm <577269746520> Tj 1 0 0 1 118.80 337.44 Tm <626F746820> Tj 1 0 0 1 143.52 337.44 Tm <736964657320> Tj 1 0 0 1 170.40 337.44 Tm <696E20> Tj 1 0 0 1 182.88 337.20 Tm <64657363656E64696E6720> Tj 1 0 0 1 239.52 337.44 Tm <706F776572732C20> Tj 1 0 0 1 281.04 336.96 Tm <65717561746520> Tj 1 0 0 1 315.36 336.96 Tm <636F656666696369656E747320> Tj 1 0 0 1 374.64 336.72 Tm <6F6620> Tj 1 0 0 1 387.60 336.72 Tm <74686520> Tj 1 0 0 1 405.84 336.72 Tm <6C696B6520> Tj 1 0 0 1 426.72 336.72 Tm <7465726D732E20> Tj 1 0 0 1 459.60 336.24 Tm <55736520> Tj 1 0 0 1 481.68 336.48 Tm <64697374726962757469766520> Tj 1 0 0 1 539.52 336.48 Tm <6C6177> Tj 0.24 Tc 1 0 0 1 48.96 323.52 Tm <6F6E20> Tj 1 0 0 1 63.84 323.76 Tm <6F757220> Tj 1 0 0 1 83.04 323.76 Tm <6571756174696F6E20> Tj 1 0 0 1 127.20 323.52 Tm <66726F6D20> Tj 1 0 0 1 153.60 323.52 Tm <7374657020> Tj 1 0 0 1 176.16 323.52 Tm <332C20> Tj 1 0 0 1 188.16 323.28 Tm <616E6420> Tj 1 0 0 1 208.32 323.28 Tm <7468656E20> Tj 1 0 0 1 232.32 323.76 Tm <67726F757020> Tj 1 0 0 1 263.28 323.04 Tm <6C696B6520> Tj 1 0 0 1 284.40 323.04 Tm <7465726D20> Tj 1 0 0 1 309.36 323.04 Tm <746F6765746865722E> Tj 1.50 Tc 1 0 0 1 231.84 308.88 Tm <782B313420> Tj 1 0 0 1 263.28 310.56 Tm <3D20> Tj 1 0 0 1 273.36 309.36 Tm <28412B20> Tj 1 0 0 1 296.88 308.64 Tm <4229782B2832412D3438292C> Tj 1.50 Tc 1 0 0 1 277.92 291.12 Tm <5B6C3D412B42> Tj 1.44 Tc /F3 15.0 Tf 1 0 0 1 265.44 273.36 Tm <222720> Tj 1 0 0 1 275.52 271.44 Tm <31746F20> Tj 1 0 0 1 294.72 274.80 Tm <3D32412E3620> Tj 1 0 0 1 330.00 273.36 Tm <3442> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 48.96 257.28 Tm <5374657020> Tj 1 0 0 1 73.44 257.28 Tm <353A20> Tj 1 0 0 1 88.80 257.04 Tm <536F6C766520> Tj 1 0 0 1 119.04 257.28 Tm <666F7220> Tj 1 0 0 1 135.60 257.28 Tm <74686520> Tj 1 0 0 1 153.12 257.28 Tm <726573756C74696E6720> Tj 1 0 0 1 198.96 257.04 Tm <6C696E65617220> Tj 1 0 0 1 229.20 256.80 Tm <73797374656D20> Tj 1 0 0 1 265.44 256.80 Tm <666F7220> Tj 1 0 0 1 282.48 256.56 Tm <652C20> Tj 1 0 0 1 297.12 256.56 Tm <422C20> Tj 1 0 0 1 311.52 256.32 Tm <432C20> Tj 1 0 0 1 325.68 256.56 Tm <6574632E> Tj 0.24 Tc 1 0 0 1 144.24 243.12 Tm <5573696E6720> Tj 1 0 0 1 175.68 243.12 Tm <737562737469747574696F6E2C20> Tj 1 0 0 1 237.36 243.12 Tm <776520> Tj 1 0 0 1 254.64 243.36 Tm <67657420> Tj 1 0 0 1 272.40 242.88 Tm <4120> Tj 1 0 0 1 284.40 245.52 Tm <3D20> Tj 1 0 0 1 294.48 242.88 Tm <3320> Tj 1 0 0 1 303.36 242.64 Tm <616E6420> Tj 1 0 0 1 323.52 242.64 Tm <4220> Tj 1 0 0 1 334.56 245.28 Tm <3D20> Tj 1 0 0 1 344.40 245.04 Tm <2D322E20> Tj 1 0 0 1 360.72 243.12 Tm <28436865636B20> Tj 1 0 0 1 398.64 242.64 Tm <7468697320> Tj 1 0 0 1 419.28 243.12 Tm <796F757273656C66272129> Tj 0.24 Tc 1 0 0 1 48.96 229.44 Tm <5374657020> Tj 1 0 0 1 73.20 229.68 Tm <363A20> Tj 1 0 0 1 85.68 229.44 Tm <5375627374697475746520> Tj 1 0 0 1 136.08 229.44 Tm <74686520> Tj 1 0 0 1 154.08 229.44 Tm <76616C756520> Tj 1 0 0 1 183.36 229.20 Tm <666F7220> Tj 1 0 0 1 200.16 229.20 Tm <4120> Tj 1 0 0 1 212.16 229.20 Tm <616E6420> Tj 1 0 0 1 232.08 229.20 Tm <4220> Tj 1 0 0 1 243.12 229.20 Tm <696E746F20> Tj 1 0 0 1 265.20 228.96 Tm <7374657020> Tj 1 0 0 1 287.76 228.72 Tm <6F6E65> Tj 1.50 Tc 1 0 0 1 191.76 206.88 Tm <2E2E2E782B74342D20> Tj 1 0 0 1 289.20 214.32 Tm <412A425F335F32> Tj 1.50 Tc 1 0 0 1 202.80 197.52 Tm <28782D342928782B322920> Tj 1 0 0 1 281.04 196.80 Tm <782D3420> Tj 1 0 0 1 317.76 196.80 Tm <782B3220> Tj 1 0 0 1 356.40 196.56 Tm <782D3420> Tj 1 0 0 1 393.12 196.56 Tm <782B32> Tj 0.24 Tc 1 0 0 1 48.48 180.96 Tm <576520> Tj 1 0 0 1 67.92 180.96 Tm <6E6F7720> Tj 1 0 0 1 91.68 180.96 Tm <6861766520> Tj 1 0 0 1 117.36 180.72 Tm <6F757220> Tj 1 0 0 1 135.84 181.20 Tm <7061727469616C20> Tj 1 0 0 1 170.40 180.72 Tm <6672616374696F6E20> Tj 1 0 0 1 210.72 180.48 Tm <6465636F6D706F736974696F6E2120> Tj 1 0 0 1 288.96 180.00 Tm <4F757220> Tj 1 0 0 1 310.32 180.24 Tm <6F726967696E616C20> Tj 1 0 0 1 351.12 180.00 Tm <66616374696F6E20> Tj 1 0 0 1 387.60 180.00 Tm <68617320> Tj 1 0 0 1 406.80 180.00 Tm <6265656E20> Tj 1 0 0 1 433.20 179.76 Tm <73706C697420> Tj 1 0 0 1 456.72 179.76 Tm <696E746F20> Tj 1 0 0 1 478.56 179.76 Tm <74776F20> Tj 1 0 0 1 500.16 179.76 Tm <736D616C6C6572> Tj 0.24 Tc 1 0 0 1 48.48 167.04 Tm <6672616374696F6E732E> Tj 0.24 Tc 1 0 0 1 48.24 139.20 Tm <546865726520> Tj 1 0 0 1 79.20 139.20 Tm <697320> Tj 1 0 0 1 90.48 139.20 Tm <616E20> Tj 1 0 0 1 105.12 139.20 Tm <616C7465726E61746520> Tj 1 0 0 1 148.56 138.96 Tm <6D6574686F6420> Tj 1 0 0 1 187.92 138.96 Tm <776520> Tj 1 0 0 1 205.20 138.72 Tm <636F756C6420> Tj 1 0 0 1 234.72 138.48 Tm <75736520> Tj 1 0 0 1 253.68 138.48 Tm <696E20> Tj 1 0 0 1 265.92 138.72 Tm <706C61636520> Tj 1 0 0 1 294.72 138.24 Tm <6F6620> Tj 1 0 0 1 308.16 138.24 Tm <737465707320> Tj 1 0 0 1 334.80 138.24 Tm <3420> Tj 1 0 0 1 344.16 138.24 Tm <616E6420> Tj 1 0 0 1 364.80 138.00 Tm <352E20> Tj 1 0 0 1 379.68 138.24 Tm <576520> Tj 1 0 0 1 399.60 138.00 Tm <63616E20> Tj 1 0 0 1 419.76 137.76 Tm <7375627374697475746520> Tj 1 0 0 1 468.72 138.00 Tm <76616C75657320> Tj 1 0 0 1 503.04 137.76 Tm <6F667820> Tj 1 0 0 1 523.68 137.76 Tm <74686174> Tj 0.24 Tc 1 0 0 1 48.24 125.28 Tm <6D616B6520> Tj 1 0 0 1 77.28 125.28 Tm <756E6B6E6F776E20> Tj 1 0 0 1 124.80 124.80 Tm <636F656666696369656E747320> Tj 1 0 0 1 184.32 124.56 Tm <6469736170706561722E20> Tj 1 0 0 1 236.40 124.32 Tm <496E20> Tj 1 0 0 1 249.36 124.32 Tm <6F757220> Tj 1 0 0 1 268.56 124.32 Tm <6578616D706C652C> Tj 0.24 Tc 1 0 0 1 48.48 110.64 Tm <537465707320> Tj 1 0 0 1 77.28 110.88 Tm <3420> Tj 1 0 0 1 86.40 110.64 Tm <616E6420> Tj 1 0 0 1 107.04 110.64 Tm <3528616C74656D617465293A20> Tj 1 0 0 1 168.24 110.40 Tm <506C756720> Tj 1 0 0 1 193.68 110.40 Tm <696E20> Tj 1 0 0 1 205.92 112.80 Tm <2D3220> Tj 1 0 0 1 220.08 110.16 Tm <666F7220> Tj 1 0 0 1 234.96 110.16 Tm <782C20> Tj 1 0 0 1 247.20 110.16 Tm <696E20> Tj 1 0 0 1 259.44 109.92 Tm <6F7264657220> Tj 1 0 0 1 288.00 109.92 Tm <746F20> Tj 1 0 0 1 300.24 109.92 Tm <6D616B6520> Tj 1 0 0 1 329.52 109.68 Tm <74686520> Tj 1 0 0 1 347.28 109.68 Tm <7465726D20> Tj 1 0 0 1 374.64 110.16 Tm <287820> Tj 1 0 0 1 387.12 109.68 Tm <2B20> Tj 1 0 0 1 396.48 109.68 Tm <32294120> Tj 1 0 0 1 420.48 109.44 Tm <6469736170706561722E20> Tj 1 0 0 1 472.80 109.44 Tm <5468697320> Tj 1 0 0 1 497.52 109.92 Tm <676976657320> Tj 1 0 0 1 525.36 109.20 Tm <7573> Tj 0.72 Tc 1 0 0 1 219.36 94.08 Tm <2E272E20> Tj 1 0 0 1 229.44 94.56 Tm <282D322920> Tj 1 0 0 1 252.24 93.84 Tm <2B723420> Tj 1 0 0 1 275.28 95.52 Tm <3D20> Tj 1 0 0 1 285.60 94.32 Tm <282D3220> Tj 1 0 0 1 304.32 93.84 Tm <2B20> Tj 1 0 0 1 313.68 93.60 Tm <32294120> Tj 1 0 0 1 334.32 93.84 Tm <2B20> Tj 1 0 0 1 343.20 94.08 Tm <282D3220> Tj 1 0 0 1 362.40 96.24 Tm <2D20> Tj 1 0 0 1 371.52 93.60 Tm <342942> Tj 1.20 Tc 1 0 0 1 277.20 77.76 Tm <2E272E313220> Tj 1 0 0 1 300.96 79.20 Tm <3D20> Tj 1 0 0 1 312.00 80.16 Tm <2D3638> Tj 1.50 Tc 1 0 0 1 282.72 62.64 Tm <3A2E383D2D32> Tj ET endstream endobj 4 0 obj 16054 endobj 5 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 6 0 obj << /Length 7 0 R >> stream Partial Fraction DecompositionMany times, it is possible to split a complicated rational expression into a sum of smaller, simplerexpressions. This sum is called the partialfraction decomposition of the original rational expression.Setting up the decomposition depends on the factors of the denominator, but.the process of solving thedecomposition is the same.In general, the decomposition will have one fraction for each factor in the denominator. If the factor islinear, the numerator of the fraction is just an unknown constant, like A. If the factor is quadractic,thenumerator will have a linear form, like A"r+B.To solve a decomposition, we solve for the unknown numerators. Let's look at a simple example.Ex. Find the partial fraction decomposition of . *I)o ^, .^ (x-4)(x+2)Step l: Set up the partial fraction decomposition with the unknown constants A, B, C, etc, in thenumerator of the decomposition.,', *t!)a =,= A.* B=,andourgoalistofindAandB.(x-4)(x+2) x-4 x+2'Step 2: Multiply both sides of the resulting equation by the least common denominator..'.(x- 4)(x+zt Uffi=(x- 4)(x+zx**4tStep 3: Simplify both sides of the equation. Every denominator should cancel out. :. @24)@.d) dh = (r,-4)(x . r) A+ (x - 4)O Ft) & :. x+14 - (x+ 2)A+@-gBStep 4: Write both sides in descending powers, equate coefficients of the like terms. Use distributive lawon our equation from step 3, and then group like term together.x+14 = (A+ B)x+(2A-48),[l=A+B"' 1to =2A.6 4BStep 5: Solve for the resulting linear system for e, B, C, etc.Using substitution, we get A = 3 and B = -2. (Check this yourself'!)Step 6: Substitute the value for A and B into step one...x+t4- A*B_3_2(x-4)(x+2) x-4 x+2 x-4 x+2We now have our partial fraction decomposition! Our original faction has been split into two smallerfractions.There is an alternate method we could use in place of steps 4 and 5. We can substitute values ofx thatmake unknown coefficients disappear. In our example,Steps 4 and 5(altemate): Plug in -2 for x, in order to make the term (x + 2)A disappear. This gives us.'. (-2) +r4 = (-2 + 2)A + (-2 - 4)B.'.12 = -68:.8=-2 endstream endobj 7 0 obj 2107 endobj 8 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 9 0 obj << /Length 44 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj2 Do Q endstream endobj 10 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj2 2 0 R >> /Subtype 6 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 8 0 R 9 0 R 3 0 R ] >> endobj 13 0 obj << /Type /XObject /Subtype /Image /Width 2551 /Height 3299 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /Columns 2551 >> /Length 46348 >> stream K{'?_]~mV`/MoվJ^w >JkBEu {?_Q}~Za_u_K z__z_K_Z֗k_KKoki?z$/WJo{onx[K\ s鄝[W޷^4p CJ8YAwA=6C:iV_ _z޾H6pATӈM1WZzy 0L-aUM &6!d 54a$ӹ J   Ma& PA)MBDDDD20A*0K 4 PAD4 <Rml$!DD2{B OJe )QZ##"6HEW ^M0AepmP(P[+yBARM0Ljӿ2SZqroJPNfqQeK'tUNБJeqUIsx}% !`hAa] `U2ꪒJcv~2S??%n`ozAAؚ3Fpq3oU[z7'NuDkޘ(&01DF'ggH[& }}u?&J_ q}"C v}>{T8d-'j>Ҫiv  úIAWV*=*C~kaaMp"y0@/D;D:`֘B&m'zֿ_j K_%n W.i6 =;_E 6wD˚EnW>}{akБ #?Z"A6oֿO Woo|$H0~NxK`ҿO}/G#ťzpwMdsޗ׫]^_k$h/߿_(ַ{kv?OzV]{k8Wa_}z HAսUU !rJi~ʴaZv7DR_wV__~?!Q0(xD_/a˯/U~a[]{Wџ[MߺOO$$=v*%a2>^iu[^ҿjK솒b~H5M4Mi#Bomsin^;[4KKo"Wh0 i,'|;M?mQ;m0a/&=zMz ÅdpIni8h:6D<ᦕ0W-¾8-Ȱd#ǰ< oo"ǯ@8dpK/ۥl zБ{GDaI4L֚8AInA@!Gy* 8n8!IiDDDhhDDDN  &ܫmV„‚V&mA":iA( a=A\V$DDD'Lxa0ЎVT!XM\DDDDEH"A, 7&aA}jDDDF6VqJIW $lT0h:>b" &A0 "?*ge;$"IP`V<&K;e*AAz$Jpaz F7 ?^|D_Z_X@ΑG3!22:گ ø0 ag<2%|BqJDj:ZDt5[]AtӼ CZPL'v f f3ff<` <|FI :/迫t09'|Z#0F aa~xޱz@`AwpCyX-4 ־>7O|a];xN F%kwkJ_j^i7;; ՂE2.2vNGh4cpҋ>OM }v]ˢ I_uz,@~-O[vJ[ɉ/B_ᆙ #]Wmb"/a֪߯d<6JwF^7>k+߮w~Ϡ~^W׫[uҴ/zݺoki^_9dȶQv}k[V?iD8i-1-+ҵ^ 7 &WAzm*{'I˫a[h:j MMB(M@^\ǧiv}cMK'GTMl Aئ!Ah5N);بm&AA8 xa>fL/K wm2,Ia0p 7iQ}\DDD[a#0 m?_DDDgԠiI0O[aT DDDDILtAm$DDDDv % #[C C t4DFD+2Fڕ+R !P dUv=PL \&6* ;gtP{P Xta?ਖ਼.*{rq'&Ob&֋x.A23 M$2[\7Ifv }(.a/=~3f˲tKȧGCDyS_U̴߄ 40@Ά0@"A< UqAšf.K־K XB 0L'츧A(D@=I430͌"jτ]d~)gJO6a&D4/D߬Zj0XA 0 I2LgAGЈjDAM3d:=R%BC5aSM?A‚ г03rujeFU]k2]A6EׇEƓ-m F#G1L!ڪhĬ30p@ !!]Љ9 /_OO_ӆEsO +kD' $;Ç *i ix!#D81SEaMutވaNi:/faǠАTm-A@q 0@qmϑ_u_MƷ__z*t]'J)2T%v4NHwKHa00xh8a 5`)4y؏TZ]=}o{"M"!.iL 蝹xi=ma~`O ЯE[] Cz z|?N pB0S ?Nȣ!&a6&RjGON 3:^""Ј1pLN¦auA ӴVX&SfAhi &t1Aw_FDDDa٭ kjM:v\DDDDDDTba?i"""6AI(0MJ"(RKZ&ZXCڨ0 )_ Ni&q; 5;)f@j/|- ;V7ph z$%\V!%GĬBцjfvKu@J!ŵ^0OU|0yR#Ⱥ#ix@H(-\H"!H22hw~o < AO320a/h0a <( d84eDE|AJu\&RIЭ\=F;PP` M31Bh0 04C\fGKZ/ |x(Ad˓ΉE SL&i0PxTvC SC03(HW}&'6ZD Ɖ;N-B駄: Bx@"AL$A9 Id@_zUᑺ dW#/4` A>#MD&#wB&|TjL0A@"f `POe'zCMt%]- 0LdHx:O(m$z'U xA0 2\S h"IrFQf?~׿[ـSKիMl O6MU#/C"CoO(0j%xBذO0g@\ BO0A Dd&_!EaQԊp[v ޞzT-6MBA {~ >|N,*a „-4SL B PL * "P.zЭevKM82-IKtŴEIAWDaOMPM??}d1O!_/[Kt[i(N  A-:w)΃h0e'tN\. o5UBj-꾞McҦNk#u' ›4I%"w/ %_c¿z4B%KJ{ߡ/I/M /@_zpr?%!`VJ-RU}<7h/Az اJ=#}L4M\/'°IV/_{^}nM?vCE?Au ^_hEp %J UhjNo_[=W쑥C/ ;}CaRBKUOWW¯mVޭt_GK^ZRkBl"8BU/ @o[Uտ]/WM;KΊ_+n$B_7'vcs鶕vǩuJ +IKJ^Ϡޒ5`_ IIz"Q C]V-bBE8ݮ6ZoIz/^44Mt.y:Ⱥ$/J2[\PL1LPARzlAwkm//`+ /I&j$mz}馩I!M2tlAhxAAA&i&qm0݄BtnIU,-VKү'Ca0C iv kdX$4MӄAR h%lZva*o G{iדKDJ]2 i˜u9O&\h0a0ZNii4C iIn"SҶ*~M8]//]K}.x@ """ R˿&8\0h0460Zk % GP҆®R0RzDDC(ȘM4L[TUa4.Ia8inBL$&"JS=Cal]].zuY`!ea{"' $dGIHG]D*jB بJ؏5J: 2=IM—04L aPa$a0a0a d @b 6/A/DDDDDDD( &L&d": *h 8i\M  a4LD&EUkt!աU1z?N`a8MDGDhXD@St]^  QGdSN"ZY҄- NCmtWK~EuW߯IaV[ZMjKWq ߿k/((O_^]TKm_#>H_wO}u~ $AD;1e0Yұ66 iqJ+Rh? 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384.48 405.60 Tm <787A20> Tj 1 0 0 1 454.80 414.96 Tm <2D2720> Tj 1 0 0 1 469.68 405.36 Tm <783220> Tj 1 0 0 1 482.88 405.12 Tm <2B6C> Tj 0.24 Tc 1 0 0 1 47.28 390.24 Tm <5374657020> Tj 1 0 0 1 71.52 390.24 Tm <333A20> Tj 1 0 0 1 84.00 390.00 Tm <53696D706C69667520> Tj 1 0 0 1 128.64 390.00 Tm <626F746820> Tj 1 0 0 1 153.84 390.00 Tm <736964657320> Tj 1 0 0 1 180.72 390.00 Tm <6F6620> Tj 1 0 0 1 193.68 390.00 Tm <74686520> Tj 1 0 0 1 211.92 390.00 Tm <6571756174696F6E2E20> Tj 1 0 0 1 259.68 389.52 Tm <457665727920> Tj 1 0 0 1 291.36 389.52 Tm <64656E6F6D696E61746F7220> Tj 1 0 0 1 355.92 389.04 Tm <73686F756C6420> Tj 1 0 0 1 390.96 389.04 Tm <63616E63656C20> Tj 1 0 0 1 424.80 388.80 Tm <6F75742E> Tj 0.72 Tc 1 0 0 1 170.16 373.92 Tm <787420> Tj 1 0 0 1 183.12 376.32 Tm <2D20> Tj 1 0 0 1 192.00 373.68 Tm <782720> Tj 1 0 0 1 205.20 373.68 Tm <2B20> Tj 1 0 0 1 214.56 373.68 Tm <347820> Tj 1 0 0 1 229.20 376.32 Tm <2D6C20> Tj 1 0 0 1 245.04 375.12 Tm <3D20> Tj 1 0 0 1 254.88 373.92 Tm <28782928787A20> Tj 1 0 0 1 286.32 373.44 Tm <2B6C29412B20> Tj 1 0 0 1 322.80 373.68 Tm <28787A20> Tj 1 0 0 1 340.32 372.96 Tm <2B6C294220> Tj 1 0 0 1 367.44 372.96 Tm <2B20> Tj 1 0 0 1 376.80 373.44 Tm <2878277A2928437820> Tj 1 0 0 1 415.68 372.72 Tm <2B20> Tj 1 0 0 1 424.80 372.48 Tm <4429> Tj 0.24 Tc 1 0 0 1 47.04 357.84 Tm <5374657020> Tj 1 0 0 1 71.04 358.08 Tm <343A20> Tj 1 0 0 1 83.52 358.08 Tm <577269746520> Tj 1 0 0 1 113.76 357.84 Tm <626F746820> Tj 1 0 0 1 138.96 357.84 Tm <736964657320> Tj 1 0 0 1 166.08 357.84 Tm <696E20> Tj 1 0 0 1 178.32 357.84 Tm <64657363656E64696E6720> Tj 1 0 0 1 235.44 357.84 Tm <706F776572732C20> Tj 1 0 0 1 276.72 357.36 Tm <65717561746520> Tj 1 0 0 1 311.28 357.12 Tm <636F656666696369656E747320> Tj 1 0 0 1 370.80 356.88 Tm <6F6620> Tj 1 0 0 1 383.76 356.88 Tm <74686520> Tj 1 0 0 1 402.00 356.88 Tm <6C696B6520> Tj 1 0 0 1 422.88 356.64 Tm <7465726D732E20> Tj 1 0 0 1 456.24 356.64 Tm <576520> Tj 1 0 0 1 475.92 356.64 Tm <77696C6C20> Tj 1 0 0 1 497.76 356.40 Tm <6861766520> Tj 1 0 0 1 523.20 356.40 Tm <746F20> Tj 1 0 0 1 535.68 356.40 Tm <757365> Tj 0.24 Tc 1 0 0 1 46.80 344.16 Tm <64697374726962757469766520> Tj 1 0 0 1 104.88 343.92 Tm <6C617720> Tj 1 0 0 1 125.04 343.92 Tm <746F20> Tj 1 0 0 1 138.00 343.68 Tm <73696D706C69667920> Tj 1 0 0 1 180.72 343.68 Tm <6F757220> Tj 1 0 0 1 200.16 343.68 Tm <6571756174696F6E20> Tj 1 0 0 1 245.28 343.20 Tm <666F726D20> Tj 1 0 0 1 271.44 343.20 Tm <7374657020> Tj 1 0 0 1 294.00 342.96 Tm <332C20> Tj 1 0 0 1 306.00 342.96 Tm <616E6420> Tj 1 0 0 1 326.40 342.96 Tm <7468656E20> Tj 1 0 0 1 350.64 343.44 Tm <676F757020> Tj 1 0 0 1 381.60 342.72 Tm <6C696B6520> Tj 1 0 0 1 402.72 342.72 Tm <7465726D7320> Tj 1 0 0 1 432.72 342.48 Tm <746F6765746865722E> Tj 1.20 Tc 1 0 0 1 119.04 327.84 Tm <787420> Tj 1 0 0 1 132.00 330.48 Tm <2D78272B34782D493D20> Tj 1 0 0 1 203.04 327.60 Tm <787428412B43292B20> Tj 1 0 0 1 261.36 327.36 Tm <7827313820> Tj 1 0 0 1 287.04 327.36 Tm <2B44292B782841292B4220> Tj 1 0 0 1 367.68 327.36 Tm <28436865636B7468697320> Tj 1 0 0 1 426.24 326.64 Tm <666F72796F757273656C666C29> Tj 1.50 Tc /F3 16.0 Tf 1 0 0 1 280.80 301.68 Tm <6C723D20> Tj 1 0 0 1 301.68 309.60 Tm <412B63> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 280.80 291.36 Tm <6C2D313D20> Tj 1 0 0 1 310.32 291.36 Tm <422B44> Tj 1.50 Tc /F3 19.0 Tf 1 0 0 1 268.32 273.36 Tm <222737303D20> Tj 1 0 0 1 303.84 273.36 Tm <6E> Tj 1.50 Tc 1 0 0 1 280.56 251.52 Tm <5B2D723D72> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 46.56 239.52 Tm <5374657020> Tj 1 0 0 1 70.80 239.52 Tm <353A20> Tj 1 0 0 1 83.28 239.28 Tm <536F6C766520> Tj 1 0 0 1 113.52 239.28 Tm <666F7220> Tj 1 0 0 1 130.56 239.52 Tm <74686520> Tj 1 0 0 1 148.08 239.28 Tm <726573756C74696E6720> Tj 1 0 0 1 193.92 239.04 Tm <6C696E65617220> Tj 1 0 0 1 224.64 239.04 Tm <73797374656D20> Tj 1 0 0 1 261.36 238.80 Tm <666F7220> Tj 1 0 0 1 277.92 238.80 Tm <412C20> Tj 1 0 0 1 293.04 238.56 Tm <422C20> Tj 1 0 0 1 307.20 238.32 Tm <432C20> Tj 1 0 0 1 321.36 238.56 Tm <6574632E> Tj 0.24 Tc 1 0 0 1 95.04 225.12 Tm <5573696E6720> Tj 1 0 0 1 127.44 225.36 Tm <737562737469747574696F6E2C20> Tj 1 0 0 1 189.84 225.12 Tm <776520> Tj 1 0 0 1 207.12 225.12 Tm <66696E6420> Tj 1 0 0 1 229.44 224.88 Tm <7468617420> Tj 1 0 0 1 251.04 224.88 Tm <413D342C20> Tj 1 0 0 1 278.16 224.64 Tm <423A2D312C20> Tj 1 0 0 1 309.36 224.16 Tm <433A2D332C20> Tj 1 0 0 1 340.80 224.40 Tm <616E6420> Tj 1 0 0 1 360.72 224.40 Tm <443D302E20> Tj 1 0 0 1 388.56 224.64 Tm <28436865636B20> Tj 1 0 0 1 426.48 223.92 Tm <7468697320> Tj 1 0 0 1 447.60 223.92 Tm <666F7220> Tj 1 0 0 1 464.88 224.40 Tm <796F757273656C662129> Tj 0.24 Tc 1 0 0 1 46.32 211.20 Tm <5374657020> Tj 1 0 0 1 70.56 211.44 Tm <363A20> Tj 1 0 0 1 83.28 211.20 Tm <5375627374697475746520> Tj 1 0 0 1 134.16 211.20 Tm <74686520> Tj 1 0 0 1 151.92 211.20 Tm <76616C756520> Tj 1 0 0 1 181.44 210.96 Tm <666F7220> Tj 1 0 0 1 198.00 210.96 Tm <74686520> Tj 1 0 0 1 216.24 210.96 Tm <636F6E7374616E747320> Tj 1 0 0 1 264.48 210.72 Tm <696E746F20> Tj 1 0 0 1 286.56 210.48 Tm <7374657020> Tj 1 0 0 1 308.64 210.24 Tm <6F6E652E> Tj 1.50 Tc 1 0 0 1 171.36 194.64 Tm <78742D78272B34782D4920> Tj 1 0 0 1 258.24 194.16 Tm <4120> Tj 1 0 0 1 280.56 193.92 Tm <4220> Tj 1 0 0 1 303.60 193.68 Tm <43782B4420> Tj 1 0 0 1 352.56 193.68 Tm <3420> Tj 1 0 0 1 371.76 196.56 Tm <2D6C20> Tj 1 0 0 1 397.44 196.32 Tm <2D33782B30> Tj 1.50 Tc /F3 4.0 Tf 1 0 0 1 247.20 188.16 Tm <3D2D492D662D3D5F745F495F> Tj 1.20 Tc /F3 11.0 Tf 1 0 0 1 184.08 177.12 Tm <782731782720> Tj 1 0 0 1 213.12 177.12 Tm <2B493120> Tj 1 0 0 1 258.72 176.88 Tm <7820> Tj 1 0 0 1 278.88 176.88 Tm <782720> Tj 1 0 0 1 307.20 176.64 Tm <787420> Tj 1 0 0 1 320.40 176.64 Tm <2B3120> Tj 1 0 0 1 352.32 176.40 Tm <7820> Tj 1 0 0 1 372.00 176.40 Tm <782720> Tj 1 0 0 1 403.20 176.16 Tm <787A20> Tj 1 0 0 1 416.64 176.40 Tm <2B6C> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 45.60 147.36 Tm <576520> Tj 1 0 0 1 65.28 147.36 Tm <6E6F7720> Tj 1 0 0 1 89.04 147.36 Tm <6861766520> Tj 1 0 0 1 114.96 147.12 Tm <6F757220> Tj 1 0 0 1 133.92 147.36 Tm <7061727469616C20> Tj 1 0 0 1 168.24 146.88 Tm <6672616374696F6E20> Tj 1 0 0 1 208.80 146.64 Tm <6465636F6D706F736974696F6E2120> Tj 1 0 0 1 290.40 146.40 Tm <496E20> Tj 1 0 0 1 302.88 146.16 Tm <6D616E7920> Tj 1 0 0 1 333.60 146.16 Tm <736974756174696F6E732C20> Tj 1 0 0 1 385.68 145.92 Tm <697420> Tj 1 0 0 1 395.28 145.92 Tm <77696C6C20> Tj 1 0 0 1 417.12 145.68 Tm <626520> Tj 1 0 0 1 432.24 145.68 Tm <65617369657220> Tj 1 0 0 1 463.20 145.68 Tm <746F20> Tj 1 0 0 1 475.68 145.68 Tm <776F726B20> Tj 1 0 0 1 503.28 145.44 Tm <7769746820> Tj 1 0 0 1 527.76 145.20 Tm <7468657365> Tj 0.24 Tc 1 0 0 1 45.84 133.44 Tm <73696D706C657220> Tj 1 0 0 1 84.96 133.20 Tm <6672616374696F6E7320> Tj 1 0 0 1 129.84 133.20 Tm <7468616E20> Tj 1 0 0 1 153.84 132.96 Tm <6F757220> Tj 1 0 0 1 173.04 132.96 Tm <6F726967696E616C2C20> Tj 1 0 0 1 216.96 132.72 Tm <636F6D706C69636174656420> Tj 1 0 0 1 279.12 132.48 Tm <6672616374696F6E2E> Tj 0.24 Tc 1 0 0 1 45.36 104.88 Tm 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90.72 Tm <7468616E20> Tj 1 0 0 1 189.60 90.48 Tm <6F6E6520> Tj 1 0 0 1 209.76 90.48 Tm <6D6574686F6420> Tj 1 0 0 1 249.60 90.24 Tm <6F6620> Tj 1 0 0 1 262.80 90.24 Tm <736F6C76696E6720> Tj 1 0 0 1 300.72 90.48 Tm <7061727469616C20> Tj 1 0 0 1 335.52 89.76 Tm <6672616374696F6E20> Tj 1 0 0 1 375.84 89.76 Tm <6465636F6D706F736974696F6E732E> Tj ET endstream endobj 15 0 obj 15695 endobj 16 0 obj << /Length 17 0 R >> stream Partial Fraction DecompositionSimilarly, we can plug in 4 for x to make the term @ - $B disappear. Solving the resulting equationgives us A:3, just like in our original method.Many people find this alternate method easier, but unfortunately, it does not always work.Let's look at a larger, more complicated example. This example has arepeated linear factor and aquadratic factor:xt - x' + 4x -l 4 The denominator has a linear factor, x, repeated twice, and a quadratic.UX]-:----:-x"(x'+I)factor, xz +1.Step l: Set up the partial fraction decomposition with the unknown constants A, B, C, etc, in thenumerator of the decomposition.xt - x' +4x-l A B Cx+ Dw|)=-,*7*T|4Sincethelinearfactorisrepeatedtwice,itneedstwofractions, one with an exponent of l, another with an exponent of 2.Step 2: Multiplyboth sides of the resulting equation by the least common denominator.(*')(r' * r) *' - :r' l l:r-r = (x2)(x2 +D 4 + (x2)7x2 + D 4 + 62y(x2 *g 9!!2' x'(x'+l) x ' xz -' x2 +lStep 3: Simplifu both sides of the equation. Every denominator should cancel out.xt - x' + 4x -l = (x)(xz +l)A+ (xz +l)B + (x'z)(Cx + D)Step 4: Write both sides in descending powers, equate coefficients of the like terms. We will have to usedistributive law to simplify our equation form step 3, and then goup like terms together.xt -x'+4x-I= xt(A+C)+ x'18 +D)+x(A)+B (Checkthis foryourselfl)lr= A+cl-1= B+D"'70= n[-r=rStep 5: Solve for the resulting linear system for A, B, C, etc.Using substitution, we find that A=4, B:-1, C:-3, and D=0. (Check this for yourself!)Step 6: Substitute the value for the constants into step one.xt-x'+4x-I A B Cx+D 4 -l -3x+0 =-I-f-=_t_I_x'1x' +I1 x x' xt +1 x x' xz +l We now have our partial fraction decomposition! In many situations, it will be easier to work with thesesimpler fractions than our original, complicated fraction.Unfortunately, this is an example where our alternate method will not get us very far. For this reason, it isimportant to know more than one method of solving partial fraction decompositions. endstream endobj 17 0 obj 2024 endobj 18 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 19 0 obj << /Length 45 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj13 Do Q endstream endobj 20 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj13 13 0 R >> /Subtype 16 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 18 0 R 19 0 R 14 0 R ] >> endobj 11 0 obj << /Kids [ 10 0 R 20 0 R ] /Count 2 /Type /Pages >> endobj 12 0 obj << /Pages 11 0 R /Type /Catalog >> endobj xref 0 21 0000000000 65535 f 0000000015 00000 n 0000000110 00000 n 0000050292 00000 n 0000066400 00000 n 0000066422 00000 n 0000066525 00000 n 0000068685 00000 n 0000068706 00000 n 0000068784 00000 n 0000068879 00000 n 0000134001 00000 n 0000134074 00000 n 0000069128 00000 n 0000115695 00000 n 0000131446 00000 n 0000131469 00000 n 0000133548 00000 n 0000133570 00000 n 0000133649 00000 n 0000133746 00000 n trailer << /Size 21 /Info 1 0 R /Root 12 0 R /ID [<91895b83821908fae3c4459ca5ccd343><91895b83821908fae3c4459ca5ccd343>] >> startxref 134129 %%EOF