# Math Help - Polynomial long division with x and y

1. ## 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?

3. ## 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.

4. ## Re: Polynomial long division with x and y

What do you define extreme point for such f(x,y)?

5. ## Re: Polynomial long division with x and y

Defn for extreme: ∂f/∂x=∂f/dy=0.

6. ## Re: Polynomial long division with x and y

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

$\{x,y\}=$

$\{0,0\}$
$\left\{-(-1)^{1/4},-(-1)^{3/4}\right\}$
$\left\{(-1)^{1/4},(-1)^{3/4}\right\}$
$\left\{-(-1)^{3/4},-(-1)^{1/4}\right\}$
$\left\{(-1)^{3/4},(-1)^{1/4}\right\}$
$\left\{-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right\}$
$\left\{-\frac{i}{\sqrt{2}},-\frac{i}{\sqrt{2}}\right\}$
$\left\{\frac{i}{\sqrt{2}},\frac{i}{\sqrt{2}}\right \}$
$\left\{\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right\}$

7. ## 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.

8. ## Re: Polynomial long division with x and y

Solve equations you specified: ∂f/∂x=∂f/dy=0.

9. ## Re: Polynomial long division with x and y

Obviously, but is there a systematic way of doing so?

10. ## 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 . . .

11. ## 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?

12. ## Re: Polynomial long division with x and y

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