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Thread: Constructing a polynomial

  1. #1
    Aug 2009

    Constructing a polynomial

    Can somebody give me a clue as to how to construct a polynomial of degree 3 from the data:
    p(xo)=f(x0), p(x1)=f(x1), p'(x0)=f'(x0), p'(x1)=f'(x1)

    using; p(x)=a0 + a1(x-x0) + a2(x-x0)^2 + a3(x-x0)^3

    I can find a0 and a1 but im a bit stuck with a2 & a3. Any help is appreciated thanks!
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  2. #2
    MHF Contributor
    Mar 2010
    I don't understand why you're using the notation f(x0), f(x1), f'(x0), f'(x1). I'm going to assume they're just constants.

    It's just a matter of doing the algebra.
    p(x)=a0 + a1(x-x0) + a2(x-x0)^2 + a3(x-x0)^3
    p'(x)=a1 + 2a2(x-x0) + 3a3(x-x0)^2

    As you've already noticed, a0 is easy, since p(x0)=a0. Also a1 is easy, since p'(x0)=a1.

    So you have
    p(x1)=a0 + a1(x1-x0) + a2(x1-x0)^2 + a3(x1-x0)^3
    p'(x1)=a1 + 2a2(x1-x0) + 3a3(x1-x0)^2
    and since you know p(x1), p'(x1), a0, a1, x0, and x1, you have two equations in the two unknowns a2 and a3.

    If you're still having trouble, please post again in this thread.

    - Hollywood
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