Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.
#color(white)(1)|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#
Step 2. Put the divisor at the left.
#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#
Step 3. Drop the first coefficient of the dividend below the division symbol.
#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#
#color(white)(1)|color(red)6#
Step 4. Multiply the result by the constant, and put the product in the next column.
#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X1)18#
#" "stackrel("—————————————)#
#color(white)(1)|color(blue)6#
Step 5. Add down the column.
#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X1)18" "color(white)(1)51color(white)(X1)132#
#" "stackrel("—————————————)#
#color(white)(1)|color(blue)6" "color(red)(color(white)(X)17#
Step 6. Repeat Steps 4 and 5 until you can go no farther.
#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X1)18" "color(white)(1)51color(white)(X1)132#
#" "stackrel("—————————————)#
#color(white)(1)|color(blue)(6)" "color(white)(X)17" "color(white)(1)44" "color(white)(1)color(red)(137)#
Answer:
#(6x^3-x^2-7x+5)/(x-3) = 6x^2+17x+44+ (137)/(x-3)#
Check:
#(x-3)(6x^2+17x+44+137/(x-3)) = (x-3)(6x^2+17x+44)+137 = 6x^3+17x^2+44x-18x^2-51x-132+137= 6x^3-x^2-7x+5#
It works!
