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Math Help - Polynomials

  1. #1
    VkL
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    Polynomials

    Find all the polynomials f(t) of degree less than or equal to 3, such that f(0) = 3, f(1) = 2, f(3) = 0 and the integral from 0 to 2 of f ' (t) dt = 4


    - Help with this problem would be appreciated a lot. Thanks.
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  2. #2
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    You have to solve a 4x4 system of linear equations.

    The polynomial is of the form ax^{3}+bx^{2}+cx+d. The first three data give you three of the equations you need in order to determine a, b, c, and d.

    Since the integral of the derivative of f between 0 and 2 is equal to f(2)-f(0), you have a fourth equation here. So, have a happy Gauss-elimination (or whatever way you call it) practice!
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  3. #3
    VkL
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    I dont get it..

    So my matrix will look something like this?

    a b c 3
    a b c 2
    a b c 0


    How do I determine what a, b and c are?
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  4. #4
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    No you plug in your values into your 3rd degree polynomial:

    <br />
ax^{3}+bx^{2}+cx+d.<br />

    so f(0)=3:

    a(0)^{3}+b(0)^{2}+c(0)+d=3

    f(1)=2

    a(1)^{3}+b(1)^{2}+c(1)+d=2

    f(3)=0

    a(3)^{3}+b(3)^{2}+c(3)+d=0

    f(2)-f(0)

    f(2)-f(0) = a(2)^{3}+b(2)^{2}+c(2)+d-(a(0)^{3}+b(0)^{2}+c(0)+d)=8a+4b+2c = 4


     \left [\begin{array}{cccc}<br />
0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 1 \\<br />
8 & 4 & 2 & 0 \\<br />
27 & 9 & 3 & 1 \end{array} \right]*\left [\begin{array}{c}<br />
a \\<br />
b \\<br />
c \\<br />
d \end{array} \right]=\left [\begin{array}{c}<br />
3 \\<br />
2 \\<br />
4 \\<br />
0 \end{array} \right]

    I put the f(3) equation in the last row btw.
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  5. #5
    VkL
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    OH, wow. Now this looks very clear. The matrix is not the hard part, its setting it up I dident understand. My first course in Linear. Thanks to both of you!!
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  6. #6
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    no problem it made me learn how to type out a matrix in Latex . . . though if you want me to show the elimination if you run into trouble with that I don't know how long that'll take me to type.
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  7. #7
    VkL
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    haha, no thats just basics. The diagnal row has to be 1's and everything zeros from what i understand.
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  8. #8
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    it's basic but its' also easy to mess up
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