A quartic polynomial has 2 distinct real roots at x = 1 and x = − 3 / 5. If the function has a y-intercept at − 1 and has f ( 2 ) = 2 and f ( 3 ) = 3. How to determine the remaining roots when two distinct real roots and y-intercept are given.

Question

A quartic polynomial has   2 distinct real roots at   x  =  1 and   x  =  −  3      /    5. If the function has a   y-intercept at   −  1 and has   f  (  2  )  =  2 and   f  (  3  )  =  3. How to determine the remaining roots when two distinct real roots and y-intercept are given.

Mackenzie Lutz · Accepted Answer

You can sum up everything you know in a system like this:      {                            f          (          1          )          =          0                                      f          (          −                      3            5                    )          =          0                                      f          (          0          )          =          −          1                                      f          (          2          )          =          2                                      f          (          3          )          =          3                        Then it&#039;s very simple to find all the coefficients of   f  (  x  )  =  a      x    4    +  b      x    3    +  c      x    2    +  d  x  +  e, because you have a   5-variable system and   5 equations, so the solution is unique! For example, from the   3rd equation we get   f  (  0  )  =  0  +  0  +  0  +  0  +  e  =  −  1  ⇒  e  =  −  1, and so on. After some calculus we obtain:      {                            a          =                      5            156                                                b          =          −                      41            78                                                c          =                      289            156                                                d          =          −                      14            39                                                e          =          −          1                        Finally we manage to find the polynomail, and the associated function:  f  (  x  )  =      5    156        x    4    −      41    78        x    3    +      289    156        x    2    −      14    39    x  −  1So all the real roots are       x    1    =  −      3    5    ,      x    2    =  1  ,      x    3    =  8  −  2      3    ,      x    4    =  8  +  2      3  .

A quartic polynomial has 2 distinct real roots at x=1 and x=−3/5. If the function has a y-intercept at−1and has f(2)=2 and f(3)=3. How to determine the remaining roots when two distinct real roots and y-intercept are given.

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