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 Polynomial Division:     more books (39)

1. Polynomial Long Division
Consequently,. An Example Long polynomial division and Factoring. Let's use polynomiallong division to rewrite. Use long polynomial division to rewrite. Answer.
http://www.sosmath.com/algebra/factor/fac01/fac01.html
##### An Example.
In this section you will learn how to rewrite a rational function such as in the form The expression is called the quotient , the expression is called the divisor and the term is called the remainder . What is special about the way the expression above is written? The remainder 28 x +30 has degree 1, and is thus less than the degree of the divisor It is always possible to rewrite a rational function in this manner: DIVISION ALGORITHM: If f x ) and are polynomials, and the degree of d x ) is less than or equal to the degree of f x ), then there exist unique polynomials q x ) and r x ), so that and so that the degree of r x ) is less than the degree of d x ). In the special case where r x )=0, we say that d x divides evenly into f x How do you do this? Let's look at our example in more detail. Write the expression in a form reminiscent of long division: First divide the leading term of the numerator polynomial by the leading term of the divisor, and write the answer 3 x on the top line: Now multiply this term 3 x by the divisor , and write the answer

2. 5 Left Or Right Polynomial Division
5 Left or right polynomial division The operator nc_divide computes the one sided quotient and remainder of two polynomials The result is a list with quotient and remainder.
http://www.uni-koeln.de/REDUCE/3.6/doc/ncpoly/node5.html

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Next: 6 Left or right polynomial reduction Up: NCPOLY: Computation in noncommutative polynomial ideals Previous: Top: REDUCE Online Documentation
##### 5 Left or right polynomial division
The operator computes the one sided quotient and remainder of two polynomials: The result is a list with quotient and remainder. The division is performed as a pseudodivision, multiplying by coefficients if necessary. The result is defined by the relation for direction left and for direction right where is an expression that does not contain any of the ideal variables, and the leading term of is lower than the leading term of according to the actual term order.
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Next: 6 Left or right polynomial reduction Up: NCPOLY: Computation in noncommutative polynomial ideals Previous: Top: REDUCE Online Documentation
REDUCE WWW Pages
maintained by Strotmann@RRz.Uni-Koeln.DE at

3. Pdiv: ----- Polynomial Division
2.7.24 pdiv polynomial division. CALLING SEQUENCE
http://www-rocq.inria.fr/scilab/doc/manual/Docu-html775.html
##### CALLING SEQUENCE
[R,Q]=pdiv(P1,P2) [Q]=pdiv(P1,P2)
##### PARAMETERS
• : polynomial matrix
• : polynomial or polynomial matrix
• R,Q : two polynomial matrices
##### DESCRIPTION
Element-wise euclidan division of the polynomial matrix by the polynomial or by the polynomial matrix Rij is the matrix of remainders, Qij is the matrix of quotients and P1ij = Qij*P2 + Qij or P1ij = Qij*P2ij + Qij
##### EXAMPLE
x=poly(0,'x'); p1=(1+x^2)*(1-x);p2=1-x; [r,q]=pdiv(p1,p2) p2*q-p1 p2=1+x; [r,q]=pdiv(p1,p2) p2*q+r-p1
ldiv gcd

4. Polynomial Long Division, Answer 2
Exercise 2. Use long polynomial division to rewrite. Answer. The divisordivides evenly into the numerator. The answer is or after
http://www.sosmath.com/algebra/factor/fac01a2/fac01a2.html
##### Exercise 2.
Use long polynomial division to rewrite
The divisor divides evenly into the numerator. The answer is: or after multiplying both sides by ( x
##### Solution.
Divide the leading term of the numerator polynomial by the leading term x of the divisor: Multiply "back": , and subtract: Divide the leading term of the bottom polynomial by the leading term x of the divisor: Multiply back: , and subtract: One more step! Divide again: Multiply "back", and subtract: The divisor divides evenly into the numerator. The answer is: or after multiplying both sides by ( x [Back] [Exercises] [Next] [Algebra] ... S.O.S MATHematics home page Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard Helmut Knaust
Fri Jun 6 13:11:33 MDT 1997

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 5. Deconvolution Or Descending Polynomial Division Deconvolution and descending polynomial division are equivalent operations. The descending polynomial b is the quotienthttp://www3.adhost.com/omatrix/manual/deconv.htm

6. Polynomial Division And Factoring
Section 6.3 polynomial division and Factoring. In the previous section,we discussed and prime polynomials. polynomial division. In order to
http://campus.northpark.edu/math/PreCalculus/Algebraic/Polynomial/Factoring/
##### Section 6.3: Polynomial Division and Factoring
In the previous section , we discussed a technique for sketching polynomials that depended on our ability to find all the factors/roots of a polynomial. This is a very difficult problem, in general, and it is complicated by the fact that the answer depends on the number system you use: rational real , or complex numbers. Using the complex numbers is very convenient, because every polynomial factors completely into complex linear factors; unfortunately, actually finding all the factors is quite challenging. On the other extreme, using only rational numbers, we have a straightforward and moderately efficient technique for finding all possible linear factors. That is because, we can quickly convert the problem to that of factoring over the integers However , not all integer rational polynomials factor completely into linear factors. Working over the real numbers is somewhere in between these two extremes. Although we cannot give explain all these difficult issues in detail (that would take several more courses in advanced mathematics!), we can discuss some specific, elementary ideas and techniques that can be quite useful in certain cases. Specifically, we will discuss:

7. Polynomial Division And Factoring: Practice Exercises
polynomial division and Factoring Practice Exercises. Here are variousExercises polynomial division and Factoring. polynomial division.
http://campus.northpark.edu/math/PreCalculus/Algebraic/Polynomial/Factoring/Exer
##### Polynomial Division and Factoring: Practice Exercises
Here are various Exercises to accompany the section Polynomial Division and Factoring
##### Polynomial Division
• Use polynomial long-division to compute the quotient, q , and remainder, r , after dividing p by d for each of the following pairs of polynomials. Check your answer by plugging your answers into the equation p d q r and simplifying.
• Divide p x x x x - 9 by d x x Divide p x x x x + 1 by d x x x Divide p x x x - 9 by d x x Pick your own polynomial, p , and a lower degree polynomial, d , and divide d into p to find the quotient, q , and remainder, r Repeat this Exercise as often as necessary until you are confident in your ability to divide polynomials
• Solution
##### Likely Factors of Rational Polynomials and Rational Roots
• Use the Rational Root Theorem to list all possible rational roots and corresponding integral, linear factors of the following polynomials.
• x x x x x x
• Solution Use our strategy for factoring rational polynomials to factor each of the following polynomials as much as possible (i.e., find as many rational roots as possible). To help you narrow your search, some values of each polynomial are already given.
• Factor p x x x x Hint x p x
Factor q x x x x Hint x p x
Factor r x x x x x Hint x p x
• Solution
##### Factoring Over the Real and Complex Numbers and Prime Polynomials
• For each of the following polynomials, factor them as much as possible over each of the following number systems:
• 8. Left Or Right Polynomial Division
Left or right polynomial division The operator nc_divide computes the one sided quotient and remainder of two polynomials The result is a list with quotient and remainder.
http://www.uni-koeln.de/REDUCE/ncpoly/section3_5.html
Next: Left or right polynomial reduction Up: NCPOLY: Computation in non-commutative polynomial ideals Previous:
##### Left or right polynomial division
The operator computes the one sided quotient and remainder of two polynomials: The result is a list with quotient and remainder. The division is performed as a pseudo-division, multiplying by coefficients if necessary. The result is defined by the relation for direction and for direction where is an expression that does not contain any of the ideal variables, and the leading term of is lower than the leading term of according to the actual term order.
Next: Left or right polynomial reduction Up: NCPOLY: Computation in non-commutative polynomial ideals Previous: Strotmann@RRz.Uni-Koeln.DE
Tue Jan 17 15:55:59 MET 1995

9. Polynomial Division Math Program
polynomial division. Created by CaS msdos version 0.5.1 Prog( MATRIX ) Defm 13 DEG NUM =16 ? AA 16= Goto 7A- B COEFS
http://www.terravista.pt/portosanto/1106/fevereiro99/polinom.htm
##### # POLYNOMIAL DIVISION
# Created by CaS msdos version 0.5.1 Prog ( MATRIX ) Defm 13:"DEG NUM
##### # Lúcio M.M. Quintal - Madeira - Portugal

10. Math Tutorials - Polynomial Division
Algebra polynomial division. This animated GIF demonstrates long division for polynomials.
##### Algebra: Polynomial Division
This animated GIF demonstrates long division for polynomials. Compare how this process is just like long division in arithmetic, but using variables instead of decimal places. The GIF will repeat every 30 seconds or so.
• Make sure there are no missing terms, insert a zero term if needed. What number, multiplied by the first term of the denominator, would equal the first term of the numerator? Multiply the denominator by that value, and subtract it from the numerator. Drop down the next term and repeat. At the end, show a remainder if any.
• ##### Synthetic Division
Synthetic division is a faster method to do certain division problems. It only works when the denominator is in the form (x + A) or (x - A). To do it you remove the variables and exponents, and just pay attention to the coefficients. Instructions for synthetic division to be added at a later date Here is the same problem solved synthetically. And at the end of the problem, you translate the coefficients back into a polynomial: x - x - 2x + 2 with remainder 3.

11. Polynomial- And Binary-Division
Polynomial and Binary-Division blue latex preamble red text in tex(red) Sponsor of the 7th LyX Developers Meeting A polynomial division writing in mathmode is possible with \underline{} , \hspace{} and use of eqnarray.
http://www.lyx.org/help/equnarray/PolDiv.php
 Polynomial- and Binary-Division blue: latex preamble red: text in tex(red) Sponsor of the 7th LyX Developers Meeting Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line Warning /home/lyx/www/www-user/help/hits2.php3 on line with equnarray A polynomial division writing in mathmode is possible with and use of eqnarray. open LyX-mathbox with alt-m-d hit ctrl-enter to produce an eqnarray-environment (two lines with three columns) type in your first line, put the equal-sign in the middle-box start second line with , lyx puts by default the closing parenthesis. for it's the same; try a value for the space

 12. Deconvolution Or Descending Polynomial Division Deconvolution or Descending polynomial divisionhttp://www.omatrix.com/manual/deconv.htm

 13. Polynomial Division The method is that of long division which we first arrange in the normal way Here we show an example of polynomial division which is needed by CRC codes to calculate the checksum which is the remainder of this division. polynomial division. x9+x5+x4+1http://www.aston.ac.uk/~blowkj/Internetworks/crc/tsld002.htm

14. Backsolving, Polynomial Division And Deconvolution
integration Backsolving, polynomial division and deconvolution. Ordinarydifferential equations often lead us to the backsolving operator.
http://sepwww.stanford.edu/sep/prof/gee/ajt/paper_html/node14.html
Next: The basic low-cut filter Up: FAMILIAR OPERATORS Previous: Causal and leaky integration
##### Backsolving, polynomial division and deconvolution
Ordinary differential equations often lead us to the backsolving operator. For example, the damped harmonic oscillator leads to a special case of equation ( ) where .There is a huge literature on finite-difference solutions of ordinary differential equations that lead to equations of this type. Rather than derive such an equation on the basis of many possible physical arrangements, we can begin from the filter transformation in ( ) but put the matrix on the other side of the equation so our transformation can be called one of inversion or backsubstitution. Let us also force the matrix to be a square matrix by truncating it with , say .To link up with applications in later chapters, I specialize to 1's on the main diagonal and insert some bands of zeros. Algebraically, this operator goes under the various names, backsolving'', polynomial division'', and deconvolution''. The leaky integration transformation ( ) is a simple example of backsolving when and a a =0. To confirm this, you need to verify that the matrices in (

15. Improved Parallel Polynomial Division And Its Extensions - Bini, Pan (ResearchIn
Improved Parallel polynomial division and Its Extensions (1992) (Make Corrections) (1 citation)
http://citeseer.nj.nec.com/bini92improved.html
 Improved Parallel Polynomial Division and Its Extensions (1992) (Make Corrections) (1 citation) Dario Bini Victor Pan September 1992 IEEE Symposium on Foundations of Computer Science Home/Search Context Related View or download: berkeley.edu/pub/tech tr92051.ps.gz Cached: PS.gz PS PDF DjVu ... Help From: berkeley.edu/techreports/1992 (more) Homepages: D.Bini V.Pan HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: (Update) Context of citations to this paper: More Recently a new, highly efficient, O(log n log n) time algorithm for finding polynomial reciprocals has been found by Bini and Pan Finally, the efficient powering circuit described in Exercise 1.8 is due to Hui Chen  1.7. Exercises 35 1.7 Exercises 1.1 Prove that... Cited by: More Fundamental Parallel Algebraic Algorithms - Newton Iteration and.. - Tate (Correct) Active bibliography (related documents): More All Improved Parallel Computations with Toeplitz-like and.. - Bini, Pan (1993) (Correct) ... (Correct) Similar documents based on text: More All Factorization of Analytic Functions by means of Koenig's.. - Dario Andrea Bini

16. Polynomial Division
band filters polynomial division. Convolution with the coefficientsb t of B(Z)=1/A(Z) is a narrowbanded filtering operation. If
http://sepwww.stanford.edu/sep/prof/pvi/zp/paper_html/node15.html
Next: Spectrum of a pole Up: DAMPED OSCILLATION Previous: Narrow-band filters
##### Polynomial division
Convolution with the coefficients b t of B Z A Z ) is a narrow-banded filtering operation. If the pole is chosen very close to the unit circle, the filter bandpass becomes very narrow, and the coefficients of B Z ) drop off very slowly. A method exists of narrow-band filtering that is much quicker than convolution with b t . This is polynomial division by A Z ). We have for the output Y Z Multiply both sides of ( ) by A Z For definiteness, let us suppose that the x t and y t vanish before t = 0. Now identify coefficients of successive powers of Z to get Let N a be the highest power of Z in A Z ). The k -th equation (where k N a ) is Solving for y k , we get Equation ( ) may be used to solve for y k once are known. Thus the solution is recursive . The value of N a is only 2, whereas N b is technically infinite and would in practice need to be approximated by a large value. So the feedback operation ( ) is much quicker than convolving with the filter B Z A Z ). A program for the task is given below. Data lengths such as

 17. Polynomial Zeros Finding Zeros of A Polynomial and polynomial division Find the rational zeros of a polynomial using the rational zeros theoremhttp://ppatten.ngc.peachnet.edu/math1113/zeros.htm

18. Polydiv - Polynomial Division
NAME. polydiv polynomial division. SYNOPSIS. standard operators parameters.DESCRIPTION. polynomial division (deconvolution), inverse to convolution.
http://www.seismo.unr.edu/ftp/pub/louie/class/706/SEP/html/lib/polydiv.html
##### NAME
polydiv - polynomial division
##### SYNOPSIS
Initializer - call polydiv_init(nd,aa) Operator - ierr=polydiv_lop(adj,add,xx,yy)
##### PARAMETERS
nd - integer
number of data points
aa - type(filter)
helix filter to perform convolution with
standard operators parameters
##### DESCRIPTION
Polynomial division (deconvolution), inverse to convolution. Requires the filter be causal with an implicit "1." at the onset.
the helix manpage the hconest manpage the helicon manpage the npolydiv manpage
##### LIBRARY

19. Npolydiv - Non Stationary Polynomial Division
NAME. npolydiv non stationary polynomial division. SYNOPSIS. DESCRIPTION. Polynomialdivision (deconvolution), inverse to convolution using space varying filter.
http://www.seismo.unr.edu/ftp/pub/louie/class/706/SEP/html/lib/npolydiv.html
##### NAME
npolydiv - non stationary polynomial division
##### SYNOPSIS
Initializer - call npolydiv_init(nd,aa) Operator - ierr=npolydiv_lop(adj,add,xx,yy)
##### PARAMETERS
nd - integer
number of data points
aa - type(nfilter)
nhelix filter to perform convolution with
standard operators parameters
##### DESCRIPTION
Polynomial division (deconvolution), inverse to convolution using space varying filter. Requires the filter be causal with an implicit "1." at the onset.