Divide x^3 + 5x^2 + x - 15 by x+3.
First, you let the coefficients of each degree to be used in the division (1, 5, 1, -15).
Then, dividing by x+3 = x - (-3) implies that you use -3 in your upper left. So, put -3, then a line (I've used a double line), and then write " 1" " 5" " 1" " -15 " to the right.
"-3 " || "1 " " 5 " " 1" " -15"
+
" " " "-----
First, bring the first 1 down to the bottom, and multiply it by the -3. Put that -3 below the 5.
"-3 " || "1 " " 5 " " 1" " -15"
+ " "" " " -3"
" " " "-----
" "" " " 1"
Then add 5+(-3) = 2 Put that under the -3
"-3 " || "1 " " 5 " " 1" " -15"
+ " "" " " -3"
" " " "-----
" " " " " 1 " " 2"
Multiply -3 xx 2 = -6 and put the -6 under the 1. Then add:
"-3 " || " 1 " " 5 " " 1 " " -15"
+ " "" " " -3 " " -6"
" " " "--------
" " " " " 1 " " 2 " " -5"
Now -3 xx -5 = 15, so we put 15 under #-15 and add::
"-3 " || "1 " " 5 " " 1 " " -15"
+ " "" ""-3 " " -6 " " 15"
" " " "--------
" "" " " 1 " " 2 " " -5 " " 0 "
The bottom row ignoring the last number gives us the coefficients of the quotient.
The last number on the bottom row is the remainder (and it is also P(-3)).
So the division gives us:
(x^3 + 5x^2 + x - 15)/(x+3) = x^2+2x-5
You can check the answer by multiplyng:
(x+3)(x^2+2x-5) to make sure we get x^3 + 5x^2 + x - 15.
(I've used Synthetic Division Formatting by Truong-Son R.)
