Show that, for every real number a ≠ 0, the equations a x 3 − x 2 − x − ( a + 1 ) = 0 and a x 2 − x − ( a + 1 ) = 0 have a common root.

Question

Show that, for every real number   a  ≠  0, the equations   a      x    3    −      x    2    −  x  −  (  a  +  1  )  =  0 and   a      x    2    −  x  −  (  a  +  1  )  =  0 have a common root.

hildiadau0o · Accepted Answer

You have almost finished the solution. But you&#039;re missing one step.Because, we must check that the solution satisfies at least one equation.You can also reach the result as follows:                                    {                                                    a                                  x                  3                                −                                  x                  2                                −                x                −                (                a                +                1                )                =                0                                                                    a                                  x                  2                                −                x                −                (                a                +                1                )                =                0                                                                                                        ⟹                                      {                                                    a                                  x                  3                                −                                  x                  2                                −                x                −                (                a                +                1                )                =                0                                                                    a                                  x                  3                                −                                  x                  2                                −                x                (                a                +                1                )                =                0                                                                                          ⟹                            x        (        a        +        1        −        1        )        =        a        +        1                                    ⟹                            x        =                              a            +            1                    a                                                        {                                                    a                                  x                  3                                −                                  x                  2                                −                x                −                (                a                +                1                )                =                0                                                                    a                                  x                  2                                −                x                −                (                a                +                1                )                =                0                                                                                                        ⟹                                      {                                                    a                                  x                  3                                −                                  x                  2                                −                x                −                (                a                +                1                )                =                0                                                                    a                                  x                  3                                −                                  x                  2                                −                x                (                a                +                1                )                =                0                                                                                          ⟹                            x        (        a        +        1        −        1        )        =        a        +        1                                    ⟹                            x        =                              a            +            1                    a                    Finally, to conclude the proof, it is sufficient to check whether the root we found satisfies the quadratic equation:            (      a      +      1              )        2              a    −            a      +      1        a    −  (  a  +  1  )  =  0.This means, you&#039;re done.

Show that, for every real number a ≠ 0 , the equations a x 3

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