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Math Help - Polynomial Function

  1. #1
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    Polynomial Function

    not sure how to do this question:

    Approximate the zeros of each function(only gonna show one, the other i'll attempt)
    f(x) = 1/4x3-2x2-2x+12

    the smaller numbers are the powers
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  2. #2
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    Quote Originally Posted by impreza02
    not sure how to do this question:

    Approximate the zeros of each function(only gonna show one, the other i'll attempt)
    f(x) = 1/4x3-2x2-2x+12

    the smaller numbers are the powers
    First sketch the function, see attachment. The function is a cubic and
    so will have no more than three zeros. From the sketch we can see that
    these will be close to  \pm 3 and 8.

    Now we will refine these estimates of the zeros using Newton Raphson
    iteration. Which is the iteration:

    <br />
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}<br />

    Using our function gives:

    <br />
x_{n+1}=x_n-\frac{0.25x_n^3-2x_n^2-2x_n+12}{0.75x_n^2-4x_n-2}<br />

    For x_0=-3, this gives:

    <br />
\begin{array}{ccc}n&x_n&x_{n+1}\\0&-3&-2.597\\1&-2.597&-2.546\\2&-2.546&-2.546\end{array}<br />

    So the root near x=-3 is \approx -2.546.

    For x_0=3, this gives:

    <br />
\begin{array}{ccc}n&x_n&x_{n+1}\\0&3&2.275\\1&2.27  5&2.281\\2&2.281&2.281\end{array}<br />

    So the root near x=3 is \approx 2.281.

    For x_0=8, this gives:

    <br />
\begin{array}{ccc}n&x_n&x_{n+1}\\0&8&8.286\\1&8.28  6&8.265\\2&8.265&8.265\end{array}<br />

    So the root near x=8 is \approx 8.265.

    RonL
    Attached Thumbnails Attached Thumbnails Polynomial Function-new-picture.png  
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