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 thecolor(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.
