To divide #3x^3+0x^2-x+4# by #x-2/3#
One Write the coefficients of #x# in the dividend inside an upside-down division symbol.
#color(white)(1)|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#
Two Put #2/3# in the divisor at the left as #x-2/3=0# gives #x=2/3#
#2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4#
#color(white)(xx)|" "color(white)(X)#
#" "stackrel("—————————————)#
Three Drop the first coefficient of the dividend below the division symbol.
#2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4#
#color(white)(X)|" "color(white)(X)#
#" "stackrel("—————————————)#
#color(white)(X)|color(white)(X)color(blue)3#
Four Multiply the result by the constant, and put the product in the next column.
#2/3|color(white)(xx)3" "color(white)(X)0color(white)(XX)-1" "" "4#
#color(white)(xx)|" "color(white)(XXx)2#
#" "stackrel("—————————————)#
#color(white)(xx)|color(white)(X)color(blue)3#
Five Add down the column.
#2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4#
#color(white)(X)|color(white)(XXXX)2#
#color(white)(1)stackrel("—————————————)#
#color(white)(X)|color(white)(X)color(blue)3color(white)(XXx)color(red)2#
Six Repeat Steps Four and Five until you can go no farther.
#2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4#
#color(white)(X)|" "color(white)(XxX)2color(white)(XXX)4/3color(white)(XxX)2/9#
#color(white)(1)stackrel("—————————————)#
#color(white)(X)|color(white)(X)color(blue)3color(white)(XxX)color(red)2color(white)(XXX)color(red)(1/3)color(white)(XxX)color(red)(4 2/9)#
Hence remainder is #4 2/9# and quotient is #3x^2+2x-1/3#
and hence #3x^3-x+4=(x-2/3)(3x^2+2x+1/3)+4 2/9#
