how can i find the remainder theorem and factor theorem to f(x)=4x^6-64x^4+x^3-19

Question

How do I find the remainder theorem and factor theorem of : f(x)=4x^6-64x^4+x^3-19??

Answer 1

 Possible Rational Roots are : ±1 ,  ± 1 , ±19 , ± 19  , ±19

                                                  ±2     ±1       ±2     ±4

 

 

     Polynomial has root of X=1 if and only if Polynomial is divisible by  X-1

 

     Do the long division to determine that:

 

             4X ⁵ +4X⁴-60X³-59X² +59X- 59                                -59X²- 19

  X - 1  l  4X⁶- 64X⁴+X³- 19                                                      -59X² -59X

              4X⁶ -4X⁵                                                                             -59X -19

                4X⁵- 64 X⁴                                                                        - -59x +59 

                 4X ⁵- 4 X⁴                                                                                  -78

                      -60X⁴+X³                     f(X) = (X-1) (4X⁵ +4X⁴ -60X3-59X² +59X -59

                      -60X⁴+60X³                   then f(1) = -78  the remainder.

                               -59X³- 19

                                59X³-59X²           f(1) = 4(1)⁶- 64(1)⁴+ (1)³ -19 =-78