To divide: # (20x^3-9x^2 +18x + 12)/(4x+3)#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
What do we need to multiply #4x# by, to get #20x^3#? We need to multiply by #5x^2#. Write that #5x^2# on the top line:
#" " " " " " "5x^2#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
Now multiply #5x^2# times the divisor, #4x+3#, to get #20x^3+15x^2# and write that under the dividend.
#" " " " " " "5x^2#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
#" "" "" "# #20x^3# #+15x^2#
#" " " " " "##-----#
Now we need to subtract #20x^3+15x^2# from the dividend. (You may find it simpler to change the signs and add. I do, so I'll do it that way)
#" " " " " " "5x^2#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
#" " " " # #color(red)(-)20x^3color(red)(-)15x^2#
#" "" "" "##-----#
#" "" "" "" "" "# #-24x^2##+18x# #+12#
Now, what do we need to multiply #4x# (the first term of the divisor) by to get #-24x^2# (the first term of the last line)? We need to multiply by #-6x#
So write #-6x# on the top line, then multiply #-6x# times the divisor #4x+3#, to get #-24x^2-18x# and write it underneath.
#" " " " " " "5x^2# #-6x#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
#" " " " # #color(red)(-)20x^3color(red)(-)15x^2#
#" "" "" "##-----#
#" "" "" "" "" "# #-24x^2##+18x# #+12#
#" "" "" "" "" "# #-24x^2##-18x#
#" " " "" "" "##------#
Now subtract (change the signs and add), to get:
#" " " " " " "5x^2# #-6x#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
#" " " " # #color(red)(-)20x^3color(red)(-)15x^2#
#" "" "" "##-----#
#" "" "" "" "" "# #-24x^2##+18x# #+12#
#" "" "" "" "" "# #color(red)(+)24x^2##color(red)(+)18x#
#" " " "" "" "##--------#
#" "" "" "" "" "" "" "" "" " # #36x# #+12#
We'll be done when the last line is #0# or has degree less than the degree of the divisor. Which has not happened yet, but we're close.
#9# times #4x# will get us #36x#, so we put the #9# on top multiply, subtract (change signs and add) to get:
#" " " " " " "5x^2# #-6x##+9#
#" " " "" "##--------#
#4x+3 )# #20x^3# #-9x^2# #+18x# #+12#
#" " " " # #color(red)(-)20x^3color(red)(-)15x^2#
#" "" "" "##-----#
#" "" "" "" "" "# #-24x^2##+18x# #+12#
#" "" "" "" "" "# #color(red)(+)24x^2##color(red)(+)18x#
#" " " "" "" "##--------#
#" "" "" "" "" "" "" "" "" " # #36x# #+12#
#" "" "" "" "" "" "" "" " # #color(red)(-)36x# #color(red)(-)27#
#" " " "" "" "" "" "##--------#
#" "" "" "" "" "" "" "" "" "" "" "##-15#
Now the last line has degree less than #1#, so we are finished.
The quotient is: #5x^2-6x+9# and the remainder is #-15#
We can write:
# (20x^3-9x^2 +18x + 12)/(4x+3) = 5x^2-6x+9 -15/(4x+3)#
IMPORTANT to understanding!
If we get a common denominator on the right and simplify we will get exactly the left side.
