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Math Help - complex quadratic polynomial

  1. #1
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    complex quadratic polynomial

    QUESTION: Find a (complex) quadratic polynomial P(z)=a+bz+cz2 which interpolates the data: (-1,i),(0,i), (6 i, 0)

    Write your answers a ,b, c in form x + iy

    My Work:
    So I have three different equations, using my x coordinate for my z values and my y coordinate as my P(z) values.
    Equations:
    i = a - b + c
    i = a
    0 = a + 6ib


    I (tried) to solve each value by matrix reduction
    __1_-1__1__i
    __1_0__0 __i
    __1_6i_-36_0

    __1__-1___ 1____i
    __0__ 1__(-6/i) _(-i/6)
    __0__-1____1___0


    (i continue to solve it until i get...):

    a=i
    b+(6i)c = -1/6
    (1+6i)c = -1/6

    if this IS right, i don't know how to actually solve the system
    i end up getting b=c,
    but if they answer has to be in x+iy how do i put it in this form??

    thanks for the help!
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  2. #2
    Super Member

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    Hello, williamb!

    Find a complex polynomial P(z)\:=\:a+bz+cz^2 which has the points: (-1,i),\;(0,i),\;(6 i, 0)

    Write your answers a ,b, c in the form x + iy
    From the given points, we have:

    . . \begin{array}{ccccccccc}P(\text{-}1) \:=\:i: & a + b(\text{-}1) + c(\text{-}1)^2 &=& i & \Rightarrow & a - b + c &=& i  & {\color{blue}[1]} \\<br />
P(0) \:=\:i: & a + b(0) + b(0^2) &=& i & \Rightarrow & a &=& i & {\color{blue}[2]}\\<br /> <br />
P(6i) \:=\:0: & a + b(6i) + c(6i)^2 & = & 0 & \Rightarrow & a + 6ib - 36c &=& 0 & {\color{blue}[3]} \end{array}


    From [2]: . \begin{array}{ccccccccc}{\color{blue}[1]}\text{ becomes:}& i - b + c &=& i & \Rightarrow & -b + c &=& 0 & {\color{blue}[4]} \\ <br /> <br />
{\color{blue}[3]}\text{ becomes:} & i + 6ib - 36c &=& 0  & \Rightarrow & 6ib - 36c &=& \text{-}i & {\color{blue}[5]}\end{array}


    \begin{array}{cccc}\text{Multiply {\color{blue}[4]} by 36:} & \text{-}36b + 36c &=& 0 \\ \text{Add {\color{blue}[5]}:} & 6ib - 36c &=& \text{-}i \end{array}

    And we have: . (\text{-}36+6i)b \:=\:\text{-}i \quad\Rightarrow\quad b \:=\:\frac{i}{36-6i} \;=\;\frac{\text{-}1+6i}{222} \;=\;\text{-}\tfrac{1}{222} + \tfrac{1}{37}i

    From [4], we have: . c \:=\:b \quad\Rightarrow\quad c \:=\:\text{-}\tfrac{1}{222}+\tfrac{1}{37}i


    Therefore, the function is: . P(x) \;=\;i + \left(-\tfrac{1}{222} + \tfrac{1}{37}i\right)x + \left(-\tfrac{1}{222} + \tfrac{1}{37}\right)x^2

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