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Homework Help: Solving Complex Numbers

  1. Jan 24, 2009 #1
    1. The problem statement, all variables and given/known data
    For a cubic polynomial P(x), with real coefficients, P(2+i)=0, P(1)=0 and P(0)=10.
    Express P(x) in the form P(x)=ax^3+bx^2+cx+d
    and solve the equation P(x)=0


    2. Relevant equations
    The conjugate factor theorem


    3. The attempt at a solution

    Using remainder theorem

    When P(2+i) = 0,

    P(2+i)=a(2+i)^3+b(2+i)^2+c(2+i)+d

    0=2a+3b+2c+d+11ai+4bi+ci


    P(1)=0
    0= a+b+c+d

    P(0)=10
    d=10

    P(2-i)=0 <--- according to the conjugate theorem

    P(2-i) =0
    0= 2a+3b+2c+d-11ai-4bi-ci


    I have trouble solving this through simultaneous equations. Is there another method?
     
  2. jcsd
  3. Jan 24, 2009 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    If P(1)=P(2+i)=P(2-i)=0, then (x-1),(x-2-i) and (x-2+i) are all factors of P(x). So, you can write
    P(x)=A(x-1)(x-2-i)(x-2+i). Then just use the fact that P(0)=10 to solve for A, and finally expand your function to get it into the desired form. :smile:
     
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