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Math Help - Find equation of cubic polynomial which passes through...

  1. #1
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    Find equation of cubic polynomial which passes through...

    Hi, I thought I knew how to solve this problem, but I don't really understand the answer I got.

    The question is as followed:

    Find the equation of a cubic polynomial which passes through (1,1), with slope -5 at this point, and through (-1,7), with slope -5 at this point.

    So, consider some things...

    p(t) = a0 + a1t + a2t^2 + a3t^3
    p`(t) = a1 + a2t + a3t^2

    p`(1) = -5, p(1) = 1, p(-1) = 7 and p`(-1) = -5


    So, I have four linear equations:

    a0 + a1 + a2 + a3 = 1, a0 - a1 + a2 - a3 = 7, a1 + a2 + a3 = -5, a1 -a2 + a3 = -5

    So, I decided to plug these equations into my ti-89 and rref it.
    1, 1, 1, 1, 1
    1, -1, 1, -1, 7
    1, 1, 1, 0, -5
    1, -1, 1, 0, -5

    but I get

    1 0 1 0 0
    0 1 0 0 0
    0 0 0 1 0
    0 0 0 0 1


    I could use some help,

    Thank you
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  2. #2
    MHF Contributor
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    Re: Find equation of cubic polynomial which passes through...

    Your p'(t) is wrong.
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  3. #3
    Member kalyanram's Avatar
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    Re: Find equation of cubic polynomial which passes through...

    Quote Originally Posted by zodiacbrave View Post
    So, I decided to plug these equations into my ti-89 and rref it.
    1, 1, 1, 1, 1
    1, -1, 1, -1, 7
    1, 1, 1, 0, -5
    1, -1, 1, 0, -5
    Matrix is incorrect the matrix has to be
     \left( \begin{matrix} 1&  1& 1& 1\\  1&  -1&  1& -1\\  0&  1&  1& 1\\  0&  1&  -1& 1\end{matrix} \right)  \left( \begin{matrix} a_0\\  a_1\\  a_2\\ a_3 \end{matrix} \right) =  \left( \begin{matrix} 1\\  7\\  -5\\ -5 \end{matrix} \right)
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