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# Find an ordered pair of constants (a,b) such that the polynomial f(x)=x^3+ax^2+(b+2)x+1 is divisible by x^2 - 1.

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Find an ordered pair of constants (a,b) such that the polynomial f(x)=x^3+ax^2+(b+2)x+1 is divisible by x^2 - 1.

waffles  Dec 4, 2017
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If this is divisible by  x^2  - 1.....it must be divisible by  (x - 1)  and ( x + 1)

So....using synthetic division

1  [   1     a         b  + 2                    1       ]

1        a  +  1          a+ b + 3

______________________________

1   a + 1     a + b + 3     a +  b + 4

-1  [   1       a           b +   2                   1          ]

-1          -a +  1               a  -  b  - 3

______________________________

1     a - 1      b - a  + 3           a  -  b  - 2

And  for 1 and -1 to be roots.....the following system must be true

a  + b  +   4   =  0

a  -  b  -   2    =  0      add  these

2a   +   2   =    0        ⇒   a  =  -1

And    a  + b  + 4  = 0

-1   + b  +  4  = 0

b + 3   =  0

b  = -3

So  (a,b)   =  ( -1, - 3)

CPhill  Dec 4, 2017

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