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Math Help - Polynomial long division with x and y

  1. #1
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    Polynomial long division with x and y

    Problem: Use polynomial long division to find the extreme point to
    f\left( x,y \right)=\left( x^{2}+y^{2} \right)\left( xy+1 \right)=x^{3}y+x^{2}+xy^{3}+y^{2}


    Attempt:
    f\left( x,y \right)=x^{3}y+x^{2}+xy^{3}+y^{2}


    \frac{\partial f}{\partial x}=3x^{2}y+2x+y^{3}


    \frac{\partial f}{\partial y}=x^{3}+3xy^{2}+2y


    I see that (0,0) is an extreme point, but I havn't learned how to use polynomial long division with more than one variable. How do I do it when I have both x and y?
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    Lightbulb Re: Polynomial long division with x and y

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    Re: Polynomial long division with x and y

    MaxJasper, does that mean 0,0 is the only point? Still: the only tools I'm allowed to have is pen, paper and—of course—polynomial long division.
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    Re: Polynomial long division with x and y

    What do you define extreme point for such f(x,y)?
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    Re: Polynomial long division with x and y

    Defn for extreme: ∂f/∂x=∂f/dy=0.
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    Lightbulb Re: Polynomial long division with x and y

    Quote Originally Posted by jacob93 View Post
    Defn for extreme: ∂f/∂x=∂f/dy=0.
    There are nine extreme points :

    \{x,y\}=

    \{0,0\}
    \left\{-(-1)^{1/4},-(-1)^{3/4}\right\}
    \left\{(-1)^{1/4},(-1)^{3/4}\right\}
    \left\{-(-1)^{3/4},-(-1)^{1/4}\right\}
    \left\{(-1)^{3/4},(-1)^{1/4}\right\}
    \left\{-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right\}
    \left\{-\frac{i}{\sqrt{2}},-\frac{i}{\sqrt{2}}\right\}
    \left\{\frac{i}{\sqrt{2}},\frac{i}{\sqrt{2}}\right  \}
    \left\{\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right\}
    Attached Thumbnails Attached Thumbnails Polynomial long division with x and y-extreme-points-contour-plot.png  
    Last edited by MaxJasper; September 14th 2012 at 11:04 AM.
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    Re: Polynomial long division with x and y

    That's good, although we don't use i in this course. My question is this: how did you get to those points? I want to learn how to find all the solution.
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  8. #8
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    Re: Polynomial long division with x and y

    Solve equations you specified: ∂f/∂x=∂f/dy=0.
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    Re: Polynomial long division with x and y

    Obviously, but is there a systematic way of doing so?
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  10. #10
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    Re: Polynomial long division with x and y

    Hello, jacob93!

    Use polynomial long division to find the extreme points to
    f\left( x,y \right)\:=\:\left( x^{2}+y^{2} \right)\left( xy+1 \right)=x^{3}y+x^{2}+xy^{3}+y^{2}

    Attempt: f\left( x,y \right)=x^{3}y+x^{2}+xy^{3}+y^{2}

    \frac{\partial f}{\partial x}=3x^{2}y+2x+y^{3}

    \frac{\partial f}{\partial y}=x^{3}+3xy^{2}+2y

    I see that (0,0) is an extreme point, but I havn't learned how to use polynomial long division
    with more than one variable. .How do I do it when I have both x and y?

    I'm not sure how long division can be applied,
    . . but I found some factoring . . . and that's all.

    We have: . \begin{array}{ccccccc}f_x &=& 3x^2y + 2x + y^3 &=& 0 & [1] \\ f_y &=& x^3 + 3xy^2 + 2y & = & 0 &[2] \end{array}

    Subtract [2] - [1]: . x^3 - y^3- 3x^2y + 3xy^2 - 2x + 2y \:=\:0

    . . . . . (x-y)(x^2+xy + y^2) - 3xy(x - y) - 2(x - y) \:=\:0

    m . . . . . . . . . . . . . (x-y)(x^2+xy+y^2-3xy - 2) \:=\:0

    n . . . . . . . . . . . . . . . . . (x-y)(x^2-2xy + y^2 - 2) \:=\:0

    . . . . . . . . . . . . . . . . . . . . . . (x-y)\left([x-y]^2-2\right) \:=\:0


    Hence we have: . \begin{Bmatrix}x-y \:=\:0 & \Rightarrow & y\:=\:x \\ (x-y)^2-2 \:=\:0 & \Rightarrow & x-y \:=\:\pm\sqrt{2} \end{Bmatrix}

    You take it from here . . .
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  11. #11
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    Re: Polynomial long division with x and y

    I saw you did [2]-[1] and I tried it myself. Couldn't do it. Looked at the rest of your post and realized I would not have thought of factoring out (x-y) which was essential. Do I lack some intuition or skill?
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  12. #12
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    Re: Polynomial long division with x and y

    Tried using the cubic rule now. \left[ 1 \right]+\left[ 2 \right]\; \Rightarrow\; \left( x+y \right)^{3}+2\left( y+x \right)\; \Rightarrow\; x=-y
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