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Fun with Polynomials PowerPoint PPT Presentation


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x 2. x 3. Fun with Polynomials. y-3x 5. -1+2y 3. 3. 1-6x-y 13. 3y 3. 6x-2xyz+5z. 5x 2 + xy 3 -6xy. 10-6x-x 10. Applying the one-variable polynomial division algorithm to several variables. -4z 2 -3xz. -x 5 + 4yz. 5xy+5x 2. -10x - 4y 7. 16x-20xz+5z. 16x-200xyz+5z. -4z 2 -3xz.

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Fun with Polynomials

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Fun with polynomials l.jpg

x2

x3

Fun with Polynomials

y-3x5

-1+2y3

3

1-6x-y13

3y3

6x-2xyz+5z

5x2 + xy3-6xy

10-6x-x10

Applying the one-variable polynomial division algorithm to several variables

-4z2-3xz

-x5 + 4yz

5xy+5x2

-10x - 4y7

16x-20xz+5z

16x-200xyz+5z

-4z2-3xz

-xy5 + 4yz


The division algorithm l.jpg

The Division Algorithm

x2

-3x+10

1

3

x+3

x3 + x + 6

5

6 8

5

x3 +3x2

Choose the leading terms

1

8

-3x2 + x +6

Proceed as usual

1 5

-3x2 - 9x

3

10x+6

remainder

10x+30

-24

3

remainder

The answer is 13

remainder

5

divisor

Algorithm terminates when we

get a difference with degree less

than that of the divisor


But what about multivariable polynomials l.jpg

But what about multivariable polynomials?

x+y x2 + 2xy + y2

What is the leading term of x+y? x2+2xy+y2 ?


Monomial orderings l.jpg

Monomial Orderings

Would like to order the monomials of x2 + 2xy + y2 .

x2 xy y2

Try ordering by degree

x2 , xy, y2 all have degree two, so need a way to break ties

Give x precedence over y

x2 precedes xy precedes y2


Back to our problem l.jpg

Back to our problem

x

+ y

x+y x2 + 2xy + y2

Identify leading terms

x2 + xy

y2 + xy

xy is the leading term here

y2 + xy

0


The ordering goes like this l.jpg

The ordering goes like this

  • First, order the variables

  • Next, order monomials by degree

  • Lastly, break ties using the order on the variables

For example, let’s order the following monomials

xy2 y3 x2y2 x2y xy3

  • First, say x precedes y

  • If we order by degree we have

xy3 x2y2 x2y y3 xy2

  • After breaking ties using the precedence of x we get

x2y2

xy3

x2y

xy2

y3


One last time l.jpg

One last time

y2 +xy x2y + 2xy2 - x2y2 + y3 -xy3

-xy

+ x

+ y

xy+y2 - x2y2 - xy3 +x2y +2xy2 +y3

Order the monomials

-x2y2 - xy3

x2y +2xy2 +y3

x2y + xy2

xy2 +y3

xy2 +y3

0

So x2y + 2xy2 - x2y2 + y3 -xy3 equals (xy+y2) (-xy+x+ ) !


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