Write the dividend expression in order of descending powers of #x# including all powers of #x# showing all coefficient values even those with coefficients of #0#
Given dividend expression becomes:
#color(white)("XXX")1x^7+(-1)x^6+1x^5+(-1)x^4+0x^3+0x^2+x^1+(-2)x^0#
Setup for synthetic division (by monic binomial)
Write the coefficients of the dividend expression as a row (line 1).
Leave a blank line or a line with just a reminder #+# sign (line 2)
Draw a separator line (optional)
Copy the#color(blue)(" value of the first coefficient")# in the column under the first coefficient to the bottom line; you may, as I have done put a prefix on this line indicating a multiplication by #color(green)("the negative of the constant term of the divisor")#.
You should have something like below (note that elements in #color(brown)("brown")# are for reference purposes only; elements in #color(green)("green")# are optional - recommended).
#{:
(,,,color(brown)(x^7),color(brown)(x^6),color(brown)(x^5),color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0)),
(color(brown)("line 1"),,"|",1,-1,+1,-1,0,0,0,-2),
(color(brown)("line 2"),+,"|",,,,,,,,),
(,,,"-----","-----","-----","-----","-----","-----","-----","-----"),
(color(brown)("line 3"),xxcolor(green)((-1)),"|",color(blue)(1),,,,,,,)
:}#
Process Steps:
1. Multiply the value in the last completed column by #color(green)("the negative of the constant term of the divisor")# and write the product on #color(brown)("line 2"#) in the next column.
2. Move to the next column and add the coefficient (from #color(brown)("line 1")# and the product produced in step 1; write the sum in this column of #color(brown)("line 3")#.
Repeat these steps until you have written a sum in the right-most column.
You should have something like:
#{:
(,,,color(brown)(x^7),color(brown)(x^6),color(brown)(x^5),color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0)),
(color(brown)("line 1"),,"|",1,-1,+1,-1,0,0,0,-2),
(color(brown)("line 2"),+,"|",,-1,+2,-3,+4,-4,+4,-4),
(,,,"-----","-----","-----","-----","-----","-----","-----","-----"),
(color(brown)("line 3"),xxcolor(green)((-1)),"|",color(blue)(1),color(blue)(-2),color(blue)(+3),color(blue)(-4),color(blue)(+4),color(blue)(-4),color(blue)(+4),color(red)(-2)),
(,,,color(brown)(x^6),color(brown)(x^5),color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0),color(brown)("R"))
:}#
The #color(red)("final sum")# is the Remainder;
the #color(blue)("preceding sums")# are the coefficients of the reduced quotient polynomial.
