How does − 1 x − 2 + 1 x − 3 become 1 2 − x − 1 3 − x I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where − 1 x − 2 + 1 x − 3 becomes 1 2 − x − 1 3 − x The equations are obviously equal, but some algebraic manipulation is done between the first step and the second step, and I can't figure out what this manipulation could be.The full breakdown comes from this solution 1 x 2 − 5 x + 6 = 1 ( x − 2 ) ( x − 3 ) = 1 − 3 − ( − 2 ) ( 1 x − 2 − 1 x − 3 ) = − 1 x − 2 + 1 x − 3 = 1 2 − x − 1 3 − x = ∑ n = 0 ∞ 1 2 n + 1 x n − ∑ n = 0 ∞ 1 3 n + 1 x n = ∑ n = 0 ∞ ( 1 2 n + 1 − 1 3 n + 1 ) x n

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How does   −      1          x      −      2        +      1          x      −      3       become       1          2      −      x        −      1          3      −      x       I&#039;m following a solution that is using a partial fraction decomposition, and I get stuck at the point where   −      1          x      −      2        +      1          x      −      3       becomes       1          2      −      x        −      1          3      −      x      The equations are obviously equal, but some algebraic manipulation is done between the first step and the second step, and I can&#039;t figure out what this manipulation could be.The full breakdown comes from this solution                                          1                                          x                2                            −              5              x              +              6                                                          =                      1                          (              x              −              2              )              (              x              −              3              )                                =                      1                          −              3              −              (              −              2              )                                            (                          1                              x                −                2                                      −                          1                              x                −                3                                      )                    =                                    −                              1                                  x                  −                  2                                            +                              1                                  x                  −                  3                                                                                                            =                                                    1                                  2                  −                  x                                            −                              1                                  3                  −                  x                                                              =                      ∑                          n              =              0                        ∞                                1                          2                              n                +                1                                                          x            n                    −                      ∑                          n              =              0                        ∞                                1                          3                              n                +                1                                                          x            n                    =                                                    ∑                                  n                  =                  0                                ∞                                            (                                  1                                      2                                          n                      +                      1                                                                      −                                  1                                      3                                          n                      +                      1                                                                      )                                            x                n

Aryanna Caldwell · Accepted Answer

Each of the terms was multiplied by             −      1              −      1      , which is really equal to 1, so it&#039;s a &quot;legal&quot; thing to do:  −            1              x        −        2              +            1              x        −        3              =  −                    (        −        1        )        1                    (        −        1        )        (        x        −        2        )              +                    (        −        1        )        1                    (        −        1        )        (        x        −        3        )              =  −                    −        1                    2        −        x              +                    −        1                    3        −        x              =            1              2        −        x              −            1              3        −        x

Frederick Kramer · Accepted Answer

I am a grade 8 student, so I may not be able to explain really well.First, I need to prove that   −      1          x      −      2        =      1          2      −      x       To prove, let&#039;s assume that &quot;x&quot; can be any number, for instance, I take x=8.So by substituting,                    −                  1                      x            −            2                                              =        −                  1                      8            −            2                                                            =        −                  1          6                    And same for this,                              1                      2            −            8                                              =                  1                      −            6                                                            =        −                  1          6                    Therefore, we have proven that     −      1          x      −      2        =      1          2      −      x      I also need to prove that       1          x      −      3        =  −      1          3      −      x      So by substituting,                               1                      8            −            3                                              =                  1          5                    and the same for this,                    −                  1                      3            −            8                                              =        −                  1                      −            5                                                            =                              −            1                                −            5                                                            =                  1          5                    Therefore, we have proven that       1          x      −      3        =  −      1          3      −      x      By why it worked? The truth is, it is just having -1÷(-1)=1 (negative×negative=positive)(And anything times 1 is the same number)So, from   −      1          x      −      2       to       1          2      −      x       , they inserted both -1 for numerator and denominator as the following below.                    −                  1                      x            −            2                                              =                              −            1                                x            −            2                                                            =                              −            1            (            −            1            )                                −            1            (            x            −            2            )                                                            =                  1                      −            x            +            2                                                            =                  1                      2            −            x                              same goes to       1          x      −      3        =  −      1          3      −      x

How does − 1 x − 2 </

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