Question 470160: What must be added to x^(3) - 3x^(2) -12x +19 so that the result is exactly divisible by x^(2) + x -6?
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! we have to find polynomials  to add to  so that the sum is divisible by  .
==> -3 and 2 are roots of the divisor  .
Now  .
By the factor theorem,
after substituting -3 into the preceding polynomial and simplifying, we must have
27a - 5b + 5c - d = 1;
after substituting 2 into the preceding polynomial and simplifying, we must have
8a +4b + 2c + d = 9;
Adding both equations, we get 7a - b + c = 2, or b = 7a + c - 2.
Substituting this into 8a +4b + 2c + d = 9, we get d = -36a - 6c + 17.
The values for a and c are free to vary.
As an example, let a = c = 1. ==> b = 7*1 + 1 - 2 = 6, and
d = -36*1 - 6*1 + 17 = -25
Then  , when added to , will be divisible by  .
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