(3x^4+5x^3-x^2+x-2)/(x-2)
There are various ways of writing the details Here's one way.
" " " "--------
x-2 ) 3x^4 +5x^3 -x^2 +x -2
What do we need to multiply the first term on the divisor (x) by to get the first term of the dividend (3x^4)? Clearly, we need to multiply by 3x^3
" " " " " " "3x^3
" " " "--------
x-2 ) 3x^4 +5x^3 -x^2 +x -2
Now multiply 3x^3 times the divisor, x-2, to get 3x^4-6x^3 and write that under the dividend.
" " " " " " "3x^3
" " " "--------
x-2 ) 3x^4 +5x^3 -x^2 +x -2
" "" "" " 3x^4 -6x^3
" " " " -----
Now we need to subtract 3x^4-6x^3 from the dividend. (You may find it simpler to change the signs and add.)
" " " " " " "" "3x^3
" " " "--------
x-2 )" " 3x^4 +5x^3 -x^2 +x -2
" " " " color(red)(-)3x^4color(red)(+)6x^3
" "" "" "-----
" "" "" "" "" " 11x^3-x^2 +x -2
Now, what do we need to multiply x (the first term of the divisor) by to get 11x^3 (the first term of the last line)? We need to multiply by 11x^2
So write 11x^2 on the top line, then multiply 11x^2 times the divisor x-2, to get 11x^3-22x^2 and write it underneath.
" " " " " " "" "3x^3 +11x^2
" " " "--------
x-2 )" " 3x^4 +5x^3 -x^2 +x -2
" " " " color(red)(-)3x^4color(red)(+)6x^3
" "" "" "-----
" "" "" "" "" " 11x^3-x^2 +x -2
" "" "" "" "" " 11x^3-22x^2
" " " "" "" "------
Now subtract (change the signs and add), to get:
" " " " " " "" "3x^3 +11x^2
" " " "--------
x-2 )" " 3x^4 +5x^3 -x^2" " +x -2
" " " " color(red)(-)3x^4color(red)(+)6x^3
" "" "" "-----
" "" "" "" "" " 11x^3-x^2" " +x -2
" "" "" "" " color(red)(-)11x^3color(red)(+)22x^2
" " " "" "" "------
" "" "" "" "" "" "" "" " 21x^2 #+x -2#
Repeat to get 21x, so we put the 9 on top multiply, subtract (change signs and add) to get:
" " " " " " "" "3x^3 +11x^2 +21x
" " " "--------
x-2 )" " 3x^4 +5x^3 -x^2" " +x -2
" " " " color(red)(-)3x^4color(red)(+)6x^3
" "" "" "-----
" "" "" "" "" " 11x^3-x^2" " +x -2
" "" "" "" " color(red)(-)11x^3color(red)(+)22x^2
" " " "" "" "------
" "" "" "" "" "" "" "" " 21x^2 #+x" "# -2
" "" "" "" "" "" "" " color(red)(-)21x^2 color(red)(+)42x
" " " "" "" "" "" "--------
" "" "" "" "" "" "" "" "" "" "" "43x -2
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.
" " " " " " "" "3x^3 +11x^2 +21x +43
" " " "--------
x-2 )" " 3x^4 +5x^3 -x^2" " +x -2
" " " " color(red)(-)3x^4color(red)(+)6x^3
" "" "" "-----
" "" "" "" "" " 11x^3-x^2" " +x -2
" "" "" "" " color(red)(-)11x^3color(red)(+)22x^2
" " " "" "" "------
" "" "" "" "" "" "" "" " 21x^2 #+x" "# -2
" "" "" "" "" "" "" " color(red)(-)21x^2 color(red)(+)42x
" " " "" "" "" "" "--------
" "" "" "" "" "" "" "" "" "" "" "43x -2
" "" "" "" "" "" "" "" "" "" "color(red)(-)43x color(red)(+)86
" " " "" "" "" "" "--------
" "" "" "" "" "" "" "" "" "" "" "" "" "" " 84
Now the last line has degree less than 1, so we are finished.
The quotient is: 3x^3+11x^2+21x+43 and the remainder is 84
We can write:
(3x^4+5x^3-x^2+x-2)/(x-2)= 3x^3+11x^2+21x+43 + 84/(x-2)
IMPORTANT to understanding what we have done:
If we get a common denominator on the right and simplify we will get exactly the left side.