synthetic division

Question

Use synthetic division to decide whether k=2, k=-1, or k=0 are zeros of
f(x)=x^4-6x^3+x^2+24x-20. If not, give the value of f(k)

Answer 1

x^4-6x^3+x^2+24x-20

 

let's try 2

 

 

2           1      -6      1      24      -20

                      2      -8    -14       20

                     

             1      -4     -7      10        0

 now divide by (x-2)

 

                x^3-4x^2-7x+10

             __________________

    x-2  / x^4-6x^3+x^2+24x-20

          /  x^4-2x^3

            ________

                    -4x^3+x^2

                    -4x^3+8x^2

                   ___________

                              -7x^2+24x

                              -7x^2+14x

                             __________

                                          10x-20

                                          10x-20

                                          _______

                                                   0

 

using x^3-4x^2-7x+10 try 1 using synthetic division

 

 

      1                  1      -4      -7      10

                                   1       -3     -10

 

                           1     -3      -10      0

 

 

divide x^3-4x^2-7x+10 by x-1

     

              x^2-3x-10

             __________________

     x-1 / x^3-4x^2-7x+10 

         /   x^3-x^2

           _________

                    -3x^2-7x

                    -3x^2+3x

                    ________

                             -10x+10

                             -10x+10

                             _______

                                        0

factor x^2-3x-10 into (x-5)(x+2)

 answer (x-2)(x-1)(x-5)(x+2) = x^4-6x^3+x^2+24x-20                       

Answer 2

Using synthetic division,

2|1 -6 1 24 -20

......2 -8 -14 20---------------------

..1 -4 -7 10|0

 

In the same way, you can see k = -1 and k = 0 are not zeros.

f(-1) = -36

f(0) = -20