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.

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