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Math Help - Newton Divided Difference Problem

  1. #1
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    Newton Divided Difference Problem

    So my problem says:
    Given the below data of polynomial P(x), determine the coefficient of all x2 in P(x) if all third-order Newton divided difference are 1

    x 0 1 2

    P(x) 2 -1 4

    What I did was find the First and second order divided differences. I think I might have done the problem incorrectly though, because I didn't use that the third order difference is 1. I was taught that the coefficient of x2 would be the second order divided difference, so I just found that using the data points.

    I got: f[x0,x1]=-3, f[x1,x2]=5, and f[x0,x1, x2]=-4
    I thought 4 would be the coefficient, but the back of the book says its 3.5. Any help?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Newton Divided Difference Problem

    Since the third order difference is constant, we know that we must have:

    P(x)=Ax^3+Bx^2+Cx+D

    Since the third order difference is 1, we have:

    P'''(x)=6A=1\:\therefore\:A=\frac{1}{6}

    Now, we are given:

    P(0)=D=2 and this results in the 2X2 system:

    B+C=-\frac{19}{6}

    4B+2C=\frac{2}{3}

    Solving this system, we find:

    B=\frac{7}{2}
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