Question

How do you divide `3x^4+5x^3-x^2+x-2` by `x-2`?

Answer 1

See in the explanation

Explanation:

We have that

`3x^4+5x^3-x^2+x-2=(3x^3+11x^2+21x+43)*(x-2)+84`

Answer 2

See the explanation section.

Explanation:

`(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^3` `color(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^3` `color(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^3` `color(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.