(Synthetic Division Formatting by Truong-Son R.)
To find #P(4)# by division, divide #x^4+x^3 + 10x^2 + 9x -6 # by #x-4#.
The Remainder Theorem to tells us that the remainder when we do the division will be equal to #P(4)#
Use synthetic division, because we've been told to. (And it is faster than long division.)
First, you let the coefficients of each degree be used in the division (#1, 1, 10, 1, -6#).
Then, dividing by #x - 4# implies that you use #4# in your upper left. So, draw the bottom and right sides of a square, put #4# inside it, and then write #"1" " 1" " 10" " 9" " -6"# to the right.
#"1" ||# #" 1" " 1" " 10" " 9" " -6"#
#+#
#"" " "##-----#
First, bring the first #1# down to the bottom, and multiply it by the #4#. Put that #4# below the second #1#.
#"4" ||# #"1 " " 1" " 10" " 9" " -6"#
#+# #"" " " "" " 4"#
#"" " """##-----#
#"" " " " 1"#
Then add it up:
#"4" ||# #"1 " " 1 " " 10" " 9" " -6"#
#+# #"" " " "" " 4"#
#"" " "##-----#
#"" " " " 1 " " 5"#
Multiply #4 xx 5# and pout the #20# under the #10#. Then add:
#"4" ||# #" 1 " " 1" " 10" " 9" " -6"#
#+# #"" " " "" " 4 " " 20"#
#"" " "##--------#
#"" " " " 1 " " 5" " 30"#
Repeat to get:
#"4" ||# #"1 " " 1" " 10" " " " 9" " " " -6"#
#+# #"" " " "" " 4" " 20 " " 120"" " "516"#
#"" " "##---------#
#"" " " " 1 " " 5" " 30 " " 129" " 510"#
The last number on the bottom row is the remainder and is also #P(4)#, so #P(4) = 510#
