#x^3+7x^2+x-18#
To carry out synthetic division, we first set:
#x+2=0=>x=-2#
This gives us the value to put in the box.
We then put the coefficients only in descending order of powers. If the polynomial has missing terms we put a zero to mark the place.
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
Next bring down the first coefficient, and put it under the bar.
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1#
Multiply the number in the division box with the number you brought down and put the result in the next column.
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-2#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1#
Add this column and put the result under the bar:
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-2#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ 5#
Multiply the number in the division box with the number you just put under the bar and put the result in the next column:
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-2 \ \ -10#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ 5#
Add the two numbers together and write the result in the bottom of the row.
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-2 \ \ -10#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ 5 \ \ \ -9#
Multiply the number in the division box with the number you brought down and put the result in the next column.
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-2 \ \ -10 \ \ \ \ \ \ \ \ \18#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ 5 \ \ \ -9#
Add this column:
# \ \ \ \ -2| \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \7 \ \ \ \ \ \ \ \1\ \ \ \ \ \ \-18#
# \ \ \ \ \ \ bar(color(white)(888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \-2 \ \ -10 \ \ \ \ \ \ \ \ \18#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ barcolor(white)(88888888888888888888)#
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ 5 \ \ \ -9 \ \ \ \ \ \ \ \ \ \0#
Write the final answer. The value in the last column is the remainder.
#x^2+5x-9+0#
Notice we write the variable in each term one power lower than the original function. This is because we are dividing by #x#
The zero remainder tells us that #(x+2)# is a factor of:
#x^3+7x^2+x-18#
And consequently #x=-2# is a root of the polynomial.
Hope this helps.
This is quite difficult to explain in just text, but if you spend some time on it, it will start to make sense.
