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When a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x-1)(x+2).

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    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) = (x-1)q2(x) + 2

    Therefore p(1) = 2.

    Then if you divide p(x) by (x-1) (x+2), the remainder will be a first degree polynomial, so we have:

    p(x) = (x-1)(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) (x-1) + 2/3 (x+2) =

    7 x - 5

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