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Factor question 2

  1. Mar 19, 2013 #1
    1. The problem statement, all variables and given/known data

    Let H(x) = x^4 + 5x^3 + 10x^2 - x - 5
    (a) Find the quotient and remainder when H(x) is divided by x^2 + 3x + 3
    (b) If H(x) + Ax + B is divisible by x^2 + 3x + 3, find the values of A and B
    (c) If H(x) + Cx^2 + Dx is divisible by x^2 + 3x + 3, find the values of C and D



    2. Relevant equations



    3. The attempt at a solution


    I am able to work out part (a)
    Quotient should be x^2 + 2x + 1 and Remainder should be -10x - 8

    For part (b),
    my attempt is
    from part (a), it is deduced that
    H(x) = Quotient x Divisor + Remainder
    H(x) = (x^2 + 3x + 3) (x^2 + 2x + 1) + (-10x - 8)
    then
    H(x) + 10x + 8 = (x^2 + 3x + 3) (x^2 + 2x + 1)

    then it is seen that A = 10 and B = 8

    but I really doubt it is correct
    because I think H(x) = (x^2 + 3x + 3) (x^2 + 2x + 1) + (-10x - 8)
    is in fact an identity, since it is true for all x, it is definitely not an equation.

    But why -10x - 8 could be 'thrown' to the left hand side
     
  2. jcsd
  3. Mar 19, 2013 #2

    SteamKing

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    It's simple algebra.

    If H(x) = Q*D + R, then it also stands to reason that
    H(x) - R = Q*D

    Nothing is being 'thrown' anywhere.
     
  4. Mar 19, 2013 #3

    SammyS

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    If [itex]\displaystyle \ H(x) = (x^2 + 3x + 3) (x^2 + 2x + 1) + (-10x - 8)\,,\ [/itex] for all x, then doesn't it follow that
    [itex]\displaystyle H(x)+(10x+8) = (x^2 + 3x + 3) (x^2 + 2x + 1) + (-10x - 8)+(10x+8)\ ? [/itex]​
     
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