Partial Fractions help!? A + C = 0 − 4 A + B − 8 C + D = 1 3 A + 16 C − 8 D = − 29 − 12 A + 3 B + 16 D = 5How do I equate the coefficients? Please provide steps an an explanation.

Question

Partial Fractions help!?  A  +  C  =  0  −  4  A  +  B  −  8  C  +  D  =  1  3  A  +  16  C  −  8  D  =  −  29  −  12  A  +  3  B  +  16  D  =  5How do I equate the coefficients? Please provide steps an an explanation.

juicilysv · Accepted Answer

Start with the first (and simplest) equation.  A  +  C  =  0 can be rearranged to give   C  =  −  ASubstitute that into the other equations:  −  4  A  +  B  −  8  C  +  D  =  1  ⇒  −  4  A  +  B  −  8  (  −  A  )  +  D  =  1  ⇒  4  A  +  B  +  D  =  1  3  A  +  16  C  −  8  D  =  −  29  ⇒  3  A  +  16  (  −  A  )  −  8  D  =  −  29  ⇒  −  13  A  −  8  D  =  −  29-12A + 3B + 16D = 5

jlo2ni5x · Accepted Answer

Form the augmented matrix.      [                                        A                                                I                                ]    =      [                            1                          0                          1                          0                          1                          0                          0                          0                                      −          4                          1                          −          8                          1                          0                          1                          0                          0                                      3                          0                          16                          −          8                          0                          0                          1                          0                                      −          12                          3                          0                          16                          0                          0                          0                          1                      ]  Gauss-Jordan reduction:Column 1            [                                                  A                                                          I                                          ]              1        =      [                            1                          0                          0                          0                                      4                          1                          0                          0                                      −          3                          0                          1                          0                                      12                          0                          0                          1                      ]              [                                                  A                                                          I                                          ]              0        =      [                            1                          0                          1                          0                          1                          0                          0                          0                                      0                          1                          −          4                          1                          4                          1                          0                          0                                      0                          0                          13                          −          8                          −          3                          0                          1                          0                                      0                          3                          12                          16                          12                          0                          0                          1                      ]  Column 2            [                                                  A                                                          I                                          ]              2        =      [                            1                          0                          0                          0                                      0                          1                          0                          0                                      0                          0                                      1            13                                    0                                      0                          −          3                          0                          1                      ]              [                                                  A                                                          I                                          ]              1        =      [                            1                          0                          1                          0                          1                          0                          0                          0                                      0                          1                          −          4                          1                          4                          1                          0                          0                                      0                          0                          1                          −                      8            13                                    −                      3            13                                    0                                      1            13                                    0                                      0                          0                          24                          13                          0                          −          3                          0                          1                      ]  Column 3            [                                                  A                                                          I                                          ]              3        =      [                            1                          0                          −          1                          0                                      0                          1                          4                          0                                      0                          0                          1                          0                                      0                          0                          −          24                          1                      ]              [                                                  A                                                          I                                          ]              2        =      [                            1                          0                          0                                      8            13                                                16            13                                    0                          −                      1            13                                    0                                      0                          1                          0                          −                      19            13                                                40            13                                    1                                      4            13                                    0                                      0                          0                          1                          −                      8            13                                    −                      3            13                                    0                                      1            13                                    0                                      0                          0                          0                          1                                      72            361                                    −                      39            361                                    −                      24            361                                                13            361                                ]  Column 4            [                                                  A                                                          I                                          ]              4        =      [                            1                          0                          0                          −                      8            13                                                0                          1                          0                                      19            13                                                0                          0                          1                                      8            13                                                0                          0                          0                          1                      ]              [                                                  A                                                          I                                          ]              3        =      [                            1                          0                          0                          0                                      400            361                                                24            361                                    −                      13            361                                    −                      8            361                                                0                          1                          0                          0                                      64            19                                                16            19                                                4            19                                                1            19                                                0                          0                          1                          0                          −                      39            361                                    −                      24            361                                                13            361                                                8            361                                                0                          0                          0                          1                                      72            361                                    −                      39            361                                    −                      24            361                                                13            361                                ]              [                                    A                                                B                                                C                                                D                              ]        =            A              −      1        b  =      1    361        [                            400                          24                          −          13                          −          8                                      1216                          304                          76                          19                                      −          39                          −          24                          13                          8                                      72                          −          39                          −          24                          13                      ]        [                            0                                      −          1                                      29                                      5                      ]    =            [                                    1                                                −            1                                                −            5                                                12                              ]

Partial Fractions help!? A+C=0 −4A+B−8C+D=1 3A+16C−8D=−29 −12A+3B+16D=5 How do I equate the coefficients? Please provide steps an an explanation.

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