Note:
#color(white)("XXX")color(magenta)5# (used below) is the value of #x# required to make the divisor, #(x-5)# equal to zero.
rows [0] and [4] are not really part of synthetic division; I include them for reference purposes only.
row [1] are the coefficients of the powers of #x# (when the expression is formed in standard notation and using #0# for any missing powers of #x#).
row [2] contains, for each column, the product of #color(magenta)5# (see above) and the value from the previous column of row [3].
row [3] contains, for each column, the sum of the values in the same column for rows [1] and [2]
The notations used in the (optional) row 4 are the powers of #x# from the (optional) row [1] reduced by 1, with the final column, marked as #color(gray)("R")#, the remainder.
#{:
([0],," | ",color(gray)(x^3),color(white)("x")color(gray)(x^2),color(white)("x")color(gray)(x^1),color(white)("x")color(gray)(x^0)),
([1],," | ",2,-11,+13,-44),
([2],ul(+color(white)("xx"))," | ",ul(color(white)("xxxx")),ul(color(white)("x")10),ul(color(white)("x")-5),ul(color(white)("x")40)),
([3],xx color(magenta)5," | ",color(white)("x")2,color(white)("x")-1,color(white)("xxx")8,color(white)("x")-4),
([4],," | ",color(gray)(x^2),color(white)("x")color(gray)(x^1),color(white)("xx")color(gray)(x^0),color(white)("xx")color(gray)R)
:}#
Normally when performing synthetic division, only rows [1], [2], and [3] are required.
