Note:
color(white)("XXX")color(magenta)5XXX5 (used below) is the value of xx required to make the divisor, (x-5)(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 xx (when the expression is formed in standard notation and using 00 for any missing powers of xx).
row [2] contains, for each column, the product of color(magenta)55 (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 xx from the (optional) row [1] reduced by 1, with the final column, marked as color(gray)("R")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.
