Editing Multiplication algorithm (section)

==Polynomial multiplication==
All the above multiplication algorithms can also be expanded to multiply [[polynomial]]s. Alternatively the [[Kronecker substitution]] technique may be used to convert the problem of multiplying polynomials into a single binary multiplication.<ref>{{citation |first1 = Joachim |last1 = von zur Gathen | author1-link = Joachim von zur Gathen |first2 = Jürgen | last2 = Gerhard |title = Modern Computer Algebra |publisher =  Cambridge University Press |year = 1999 |isbn = 978-0-521-64176-0 |pages = 243–244 |url = https://books.google.com/books?id=AE5PN5QGgvUC&pg=PA245 }}.</ref>

Long multiplication methods can be generalised to allow the multiplication of algebraic formulae:

  14ac - 3ab + 2 multiplied by ac - ab + 1

  14ac  -3ab   2
    ac   -ab   1
  ————————————————————
  14a<sup>2</sup>c<sup>2</sup>  -3a<sup>2</sup>bc   2ac
         -14a<sup>2</sup>bc         3 a<sup>2</sup>b<sup>2</sup>  -2ab
                  14ac           -3ab   2
  ———————————————————————————————————————
  14a<sup>2</sup>c<sup>2</sup> -17a<sup>2</sup>bc   16ac  3a<sup>2</sup>b<sup>2</sup>    -5ab  +2
  <nowiki>=======================================</nowiki><ref>{{cite book|last1=Castle|first1=Frank|title=Workshop Mathematics|url=https://archive.org/details/workshopmathema00castgoog|date=1900|publisher=MacMillan and Co|location=London|page=[https://archive.org/details/workshopmathema00castgoog/page/n88 74]}}</ref>

As a further example of column based multiplication, consider multiplying 23 long tons (t), 12 hundredweight (cwt) and 2 quarters (qtr) by 47.  This example uses [[avoirdupois]] measures: 1 t = 20 cwt, 1 cwt = 4 qtr.

     t    cwt  qtr
    23     12    2
                47 x
  ————————————————
   141     94   94
   940    470
    29     23
  ————————————————
  1110    587   94
  ————————————————
  1110      7    2
  <nowiki>=================</nowiki>  Answer: 1110 ton 7 cwt 2 qtr

First multiply the quarters by 47, the result 94 is written into the first workspace.  Next, multiply cwt 12*47 = (2 + 10)*47 but don't add up the partial results (94, 470) yet.  Likewise multiply 23 by 47 yielding (141, 940).  The quarters column is totaled and the result placed in the second workspace (a trivial move in this case).  94 quarters is 23 cwt and 2 qtr, so place the 2 in the answer and put the 23 in the next column left.  Now add up the three entries in the cwt column giving 587.  This is 29 t 7 cwt, so write the 7 into the answer and the 29 in the column to the left.  Now add up the tons column.  There is no adjustment to make, so the result is just copied down.

The same layout and methods can be used for any traditional measurements and non-decimal currencies such as the old British [[£sd]] system.