(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#