For what value of c will the polynomial P(x) = -2x^3 + cx^2 - 5x + 2 have the same remainder when it is divided by x - 2 and by x + 1?

We can use some polynomial "long division" to help us with this one.....

                          -2x^2          + [c- 4]x          + [2c - 13]

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x - 2     -2x^3 + cx^2            - 5x                       + 2

            -2x^3  + 4x^2

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                          (c -4)x^2      -5x

                          (c - 4)x^2    -2(c - 4)x

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                                              [2c - 13]x               + 2                                        

                                              [2c - 13]x              -2[2c - 13]

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                                                                             4c  - 24

                             -2x^2          + [c + 2]x          - [ 7 + c]

             -------------------------------------------------------------

x + 1    -2x^3  + cx^2             - 5x                      + 2

             -2x^3  -  2x^2

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                           [c + 2)x^2     -5x                                                             

                           [c + 2]x^2    + [c + 2)x

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                                                 - [7 + c] x            + 2

                                                 - [7 + c] x          - [7 + c]

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                                                                              9 + c

And it's obvious that the remainders will be equal when.....

4c - 24 = 9 + c      subtract c from both sides and add 24 to both sides

3c  =  33   .....  so .......

c = 11

That was quite a feat Chris,  

I will do it using remainder theorem.

For what value of c will the polynomial P(x) = -2x^3 + cx^2 - 5x + 2 have the same remainder when it is divided by x - 2 and by x + 1?

When P(x) is divided by x-2 the remainder will be     p(2)=-2*8+c*4-10+2 = -16+4c-8 = 4c-24

When P(x) is divided by x+1 the remainder will be    p(-1)=-2*-1+c*1+5+2=2+c+7 = 9+c

If these are the same then 

4c-24=9+c

3c=33

c=11