How do you use synthetic division to divide# (x^2+4x+1)# by #x-5#?

1 Answer
Feb 11, 2016

Answer:

#x+9+46/(x-5)#

For explanation, see below.

Explanation:

Dividing by #x-5# means that the corresponding root of #5# will be in the upper left of the synthetic division. The numbers up top will be the coefficients of #x^2+4x+1#, which is #1,4,1#.

#{:(ul5"|"," 1"," 4"," 1"," "),(ul" ",ul" ",ul" ",ul" ",+),(" "," "," ","| "," "):}#

Move the #1# in the first column down to the bottom.

#{:(ul5"|"," 1"," 4"," 1"," "),(ul" ",ul" ",ul" ",ul" ",+),(" "," 1"," ","| "," "):}#

Multiply the #1# by the #5# in the corner for a product of #5#, which goes in the next column.

#{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul" ",+),(" ","      "color(red)1," ","| "," "):}#

Add the #4# and the #5# in the new column.

#{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul" ",+),(" ","      "color(red)1," "9,"| "," "):}#

Multiply the #9# by the #5# in the corner for #45#, which goes in the next column.

#{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul("    "color(blue)(45)),+),(" ","      "color(red)1," "color(blue)9,"| "," "):}#

Add the #1# and #45#. This goes in the "remainder" spot.

#{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul("    "color(blue)(45)),+),(" ","      "color(red)1," "color(blue)9,"|46"," "):}#

The numbers #1,9# in the bottom row correlate to the coefficients of the quotient. The #46# is the numerator of the remainder quotient:

#(x^2+4x+1)/(x-5)=x+9+46/(x-5)#