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Math Help - Drawing a polynomial

  1. #1
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    Drawing a polynomial

    My question gives a polynomal with a degree of 16:

    y = 1/288 (x-3)^2 (x+2)^3 (x+1)^9 (x-4)^2

    So the polynomial crosses the x axis at point:
    3 two times
    -2 three times
    -1 nine times
    4 two times

    My question is how the graph would look like. I guessed that since there really is only 4 x-intercepts and that the rest just overlap, that it's a quartic function, therefore giving the shape M.

    But then it's a 16 degree polynomial so it doesnt make sense, is it a combination of quartics and cubics? Can someone help?
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by onenameless View Post
    My question gives a polynomal with a degree of 16:

    y = 1/288 (x-3)^2 (x+2)^3 (x+1)^9 (x-4)^2
    So the polynomial crosses the x axis at point:
    3 two times
    -2 three times
    -1 nine times
    4 two times

    My question is how the graph would look like. I guessed that since there really is only 4 x-intercepts and that the rest just overlap, that it's a quartic function, therefore giving the shape M.

    But then it's a 16 degree polynomial so it doesnt make sense, is it a combination of quartics and cubics? Can someone help?
    When we have repeated root in a polynomial a few different things can happen.

    If the root is repeated an even number of time like 3 above the graph bounces (doesn't cross the x-axis) there.

    If the root is repeated an odd number of times it will cross the x-axis there, and have a horizontal tangent line at the point. like -2 above is repeated 3 times

    The graph will have "U" like end behavior becuase the polynomial has an even degree.

    Drawing a polynomial-capture.jpg
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  3. #3
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    oh i see. so i was kind of on the right track, thanks.
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  4. #4
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    Why do even multiplicities touch but odd pass through?

    I am curious as to why roots repeated an even number of times "touch" the x-axis but those repeated an odd number of times pass through? I know using calculus that if a root has multiplicity 2 or higher then that root is also a root of the derivative.
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