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Math Help - Find a cubic polynomial P for which...

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    Find a cubic polynomial P for which...

    Apostol, 1.26.24

    Problem: Find a cubic polynomial P for which P(0) = P(-2) = 0, P(1) = 15, and the integral of P from -2 to 0 is 4/3.

    I can find a polynomial that satisfies the first 3 conditions with no problem:
    x(x + 2)(x + c)...
    (1)((1) + 2)((1) + c) = 15
    x^3 + 6x^2 + 8x

    However, I calculated the integral of x^3 + 6x^2 + 8x from -2 to 0 to be 0. Thus, I can't scale it with a constant, and I don't know of any other transformations that would allow me to preserve the intercepts while setting the area to 4/3.

    I don't need the answer -- it's in the book. But any advice on getting from here to there would be much appreciated.
    Last edited by Dementiy; April 1st 2010 at 08:35 PM.
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    Quote Originally Posted by Dementiy View Post
    Apostol, 1.26.24

    Problem: Find a cubic polynomial P for which P(0) = P(-2) = 0, P(1) = 15, and the integral of P from -2 to 0 is 4/3.

    I can find a polynomial that satisfies the first 3 conditions with no problem:
    x(x + 2)(x + c)...
    (1)((1) + 2)((1) + c) = 15
    x^3 + 6x^2 + 8x

    However, I calculated the integral of x^3 + 6x^2 + 8x from -2 to 0 to be 0. Thus, I can't scale it with a costant, and I don't know of any other transformations that would allow me to preserve the intercepts while setting the area to 4/3.

    I don't need the answer -- it's in the book. But any advice on getting from here to there would be much appreciated.
    Try setting your polynomial as

    P(x) = kx(x + 2)(x + c) instead.
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