math
posted by KnowsNothing .
When a polynomial is divided by (x+2), the remainder is 19. When the same polynomial is divided by (x1), the remainder is 2. Determine the remainder when the polynomial is divided by (x1)(x+2).

If p(x) is the polynomial, then you have:
p(x) = (x+2)q1(x)  19
for some poynomial q1(x). You see that the remainder of 19 is the value of
p(x) at x = 2. We also have:
p(x) = (x1)q2(x) + 2
Therefore p(1) = 2.
Then if you divide p(x) by (x1) (x+2), the remainder will be a first degree polynomial, so we have:
p(x) = (x1)(x+2)q3(x) + r(x)
Then if you put x = 1 in here and use that p(1) = 2, you find:
r(1) = 2
Putting x = 2 and using that
p(2) = 19 yields:
r(2) = 19
These two values of r(x) fix r(x) as
r(x) is of first degree. We have:
r(x) = (19)/(3) (x1) + 2/3 (x+2) =
7 x  5
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