To divide #x^3-7x+6=x^3+0x^2-7x+6# by #x-2# (as we have to show #x=2#, a zero).
One Write the coefficients of #x# in the dividend inside an upside-down division symbol.
#color(white)(1)|color(white)(X)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#
Two As #x=2# we put #2# at the left.
#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)#
#" "stackrel("—————————————)#
Three Drop the first coefficient of the dividend below the division symbol.
#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(X)#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(red)1#
Four Multiply the result by the constant, and put the product in the next column.
#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)2#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(blue)1#
Five Add down the column.
#2|color(white)(x)1" "color(white)(x)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)2#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(blue)1color(white)(X11)color(red)2#
Six Repeat Steps Four and Five until you can go no farther.
#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)2color(white)(xxxxx)4color(white)(X)-6#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(blue)1color(white)(X11)color(red)2color(white)(XX)color(red)-3color(white)(XXX)color(red)0#
Hence, Quotient is #x^2+2x-1# and remainder is #0#.
As remainder is #0#, #x=2# is a zero of #x^3-7x+6#.
